President Trump on Saturday urged officials in Minnesota to “get tougher” on the protesters and offered greater military support, a move that would represent a significant escalation in the government’s response to the tensions. Gov. Tim Walz declined the Army’s offer to deploy military police units, but said he had activated all 13,000 of the state’s National Guard troops and warned that Saturday night’s protest could be the largest and most destructive yet.

Mayor Jacob Frey of Minneapolis urged protesters to stay off the streets Saturday night. “We need you to stay home tonight to be able to respond to those who are trying to loot and burn our city,” he wrote on Twitter.

The massive crowds represented a sudden departure for cities that had previously been under lockdown orders amid the coronavirus pandemic, and officials quickly found themselves shifting from one crisis to another. In Louisville, Ky., where protests have centered on the death of Breonna Taylor, a black woman who was shot dead by white police officers who entered her home in March, Mayor Greg Fischer imposed a curfew and called in the National Guard. In Los Angeles, Mayor Eric M. Garcetti also issued a curfew, a day after the police made more than 500 arrests.

In New York City, thousands of demonstrators took to the streets for a third day, gathering at marches in Harlem, Brooklyn, Queens and outside Trump Tower in Midtown Manhattan. In the late afternoon, protesters in the Flatbush section of Brooklyn confronted the police in a series of street melees, hurling empty bottles and pieces of debris at officers who responded with billy clubs and pepper spray.

Many of the hundreds marching through Center City in Philadelphia on Saturday were doing so peacefully, with fists raised and signs held aloft in anger over Mr. Floyd’s death. But there were reports of destruction as the afternoon wore on.

At least one police car caught on fire, and a large group of protesters had spray-painted a statue of former Mayor Frank Rizzo and were trying to topple it. Mr. Rizzo, a former police commissioner who died in 1991, cultivated a law-and-order image that included raiding gay clubs and once forcing Black Panthers to strip naked in the street. He remains a figure loathed by many for his harsh tactics. The current mayor, Jim Kenney, had announced plans to move the statue to a new location.

In Tallahassee, Fla., the driver of a red pickup truck struck a crowd of protesters, in a fleeting but terrifying episode that officials said did not end in serious injury. The protest there came days after the Tallahassee police fatally shot Tony McDade, a black transgender person whom the police had identified as a suspect in a stabbing.

In Minneapolis, about 300 to 400 protesters gathered outside the home of the Hennepin County Attorney, Mike Freeman, holding signs and listening to speakers, including the mothers of children killed by the police, according to a neighbor, Clare Padgett. It was the fourth day of protests outside the home of the prosecutor, who charged the police officer who pinned Mr. Floyd but has not pressed charges against other officers who were on the scene.

Minnesota’s governor activated all National Guard troops, but declined the Army’s offer to deploy military police.

Gov. Tim Walz of Minnesota activated thousands of additional National Guard troops to send to Minneapolis but declined the Army’s offer to deploy military police units, as days of protests over the death of Mr. Floyd threatened to boil over even further on Saturday.

Mr. Walz, a Democrat, acknowledged that officials had underestimated the demonstrations in Minneapolis, where despite a newly issued curfew, people burned buildings and turned the city’s streets into a smoldering battleground on Friday night. He compared the havoc to wars that Americans have fought overseas, and said he expected even more unrest on Saturday night.

“What you’ve seen in previous nights, I think, will be dwarfed by what they will do tonight,” he said.

Pentagon officials said that Defense Secretary Mark Esper and General Mark Milley, the chairman of the Joint Chiefs of Staff, spoke Friday with Mr. Walz, to express “willingness” to deploy military police units. The governor declined the offer, the officials said, and has since activated all of the state’s National Guard troops, up to 13,200.

Nonetheless, the U.S. Northern Command has put several military police units on four-hour status, which means they could be ready to deploy in four hours, as opposed to a day.

Commissioner John Harrington of the Minnesota Department of Public Safety said that there had been “tens of thousands” of people in the streets on Friday, more than any other night since Mr. Floyd’s death on Monday set off a wave of protests that have become increasingly destructive across the country.

Mayor Jacob Frey of Minneapolis, looking weary after four days of outrage in his city, pleaded with residents to go home and stop burning down the local businesses that he said were even more vital in the middle of a pandemic.

An anguished nation watches the unrest with a mix of hope and horror.

The mass demonstrations, some peaceful, some destructive, touched off an anguished debate on Saturday among commentators, pastors and scholars about the role of protest in forcing political and social change in America.

In the pages of the nation’s newspapers and on social media, some watched in horror and others with a sense of hope as protesters took to the streets following Mr. Floyd’s death.

The singer Selena Gomez said she had spent the last 24 hours “trying to process this all.”

“Nothing anyone says can take back what has happened,” she wrote on Twitter. “But we can and must all make sure to take action. Too many black lives have been taken from us for far too long.”

Shana L. Redmond, a scholar of music, race and politics at U.C.L.A.’s Herb Alpert School of Music, described receiving tearful calls from loved ones in Minneapolis with “tears running too fast to pause at sadness.”

“They are hot with rage and anger at the condition of Black people in that place and in this world,” she wrote on Twitter. “This is not a drill. This is our terrifying, murderous present. #MinneapolisRebellion.”

“I understand our frustration, rage and anger but it’s of no benefit when we hurt ourselves in the process,” Jamal H. Bryant, the pastor of New Birth Missionary Baptist Church in Lithonia, Ga., wrote on Twitter. “I urge adamantly for the fixing of this corrupt justice system and I pray profusely for the black businesses adversely affected!”

Senator Tim Scott of South Carolina, the only black Republican in the Senate, who has called for the arrest of all four officers involved the arrest of Mr. Floyd, urged calm.

“I know well the anger and frustration felt throughout communities of color right now,” he wrote on Twitter. “But violence is not the answer. Violence takes the focus off #GeorgeFloyd and the real issues at hand, and gives those who prey on division more fuel. Make your voices heard, not bricks and fire.”

Brian Merchant, an author, noted how the nation was briefly distracted on Saturday by the launch of a rocket built and operated not by NASA but SpaceX, the company founded by the billionaire Elon Musk.

“The symbolism of a billionaire-owned, for-profit space company launching astronauts high above the heads of thousands of people protesting state brutality and oppression amid a global pandemic and economic collapse should not be lost on anyone,” he wrote.

On Blavity, a website geared toward black millennials, an editorial argued that the fires in Minneapolis reflected “the rage of Black protestors fed up seeing the lives of our brothers and sisters robbed by racism.”

“We are fed up because we are forced to fight a pandemic amid a pandemic,” the editorial said. “We are being disproportionately killed by systemic and overt racism at the same time — and are expected to accept these deadly conditions.”

In The Los Angeles Times, an editorial argued that the country should focus should not on the looting but on the repeated instances of the police killing black men.

“And no, police violence does not justify the rampages that erupted in Minneapolis,” the editorial said. “But as the Rev. Martin Luther King Jr. pointed out more than a half-century ago, ‘a riot is the language of the unheard.’”

Trump blames ‘left-wing groups’ for the violence, and Barr threatens rioters with prosecution.

President Trump on Saturday blamed the unrest in cities across the country on “Antifa and other radical left-wing groups,” drawing a distinction between “peaceful protesters” and other, more violent demonstrators.

“I understand the pain that people are feeling,” Mr. Trump said, speaking from the Kennedy Space Center in Florida after observing the launch of a manned SpaceX rocket. “We support the right of peaceful protesters, and we hear their pleas. But what we are now seeing on the streets of our cities has nothing to do with justice or with peace. The memory of George Floyd is being dishonored by rioters, looters and anarchists.”

While he called George Floyd’s death “a grave tragedy,” Mr. Trump also said that residents affected by violent protests were the “main victims of this horrible, horrible situation.”

Speaking on the South Lawn of the White House earlier in the day, Mr. Trump criticized the authorities in Minnesota for allowing protests to turn violent, and offered the help of the military to contain further demonstrations.

“They have to get tougher, and by being tougher they will be honoring his memory,” the president said, adding: “When I saw the policemen running out of a police station for that police station to be abandoned and taken over, I’ve never seen anything so horrible and stupid in my life.”

His comments, paired with a series of tweets on Saturday, threatened to inflame an already tense situation that has played out in protests across the country and in front of the White House.

In one tweet, he called demonstrators who gathered at the White House on Friday night “professionally managed so-called ‘protesters’” and suggested that his supporters would march outside the White House on Saturday.

“Tonight, I understand, is MAGA NIGHT AT THE WHITE HOUSE???” he tweeted.

Asked later whether the tweet might have invited more violence, Mr. Trump demurred. “These are people that love our country,” he said of his supporters. “I have no idea if they were going to be here, I was just asking.”

“By the way,” he added, “they love African-American people, they love black people.”

Attorney General William P. Barr, who has vowed a swift federal investigation into Mr. Floyd’s death, also weighed in on Saturday, issuing a stern warning to left-wing “agitators” who he said were exploiting the protests to pursue their own goals.

The United States attorney in Minnesota is investigating the actions of the police officer who pinned Mr. Floyd to the ground for possible violations of civil rights laws or other federal crimes.

In Atlanta, protesters and curiosity-seekers visited CNN headquarters to see the damage.

After hundreds of demonstrators poured into the streets of Atlanta on Friday night, smashing windows, vandalizing a large CNN sign and clashing with police officers, a more muted crowd arrived at the scene on Saturday — some out of curiosity, and some to continue protesting.

Bianca Billups, 31, stood holding a sign for the cars passing by to see: “White Supremacy Did This.” She had been there for hours, she said, and she had also been at the protests on Friday evening.

“They think people are out here causing havoc because they can,” she said. “People drive by here and they’ll see this and think, ‘Those people are so reckless.’ No, those people are scared and they’re in survivor mode. They’re done. They’re fed up.”

Ty Harris, 41, walked up with a hat and scarf covering his face; both said, “Make America United For Once.” “I think this is the spark that’s going to change life as we know,” he said.

On the edge of Centennial Olympic Park, a series of people came up to speak into a shared microphone. “They want you to fear,” said one man who added that he had been stopped by the police some 20 times because, he said, he looked “suspicious.”

Some of the buildings near the CNN Center, the epicenter of the Friday protests, had smashed windows and were protected by fencing.

Joscie Beachum, a 27-year-old Atlanta resident looking at the damaged CNN Center, said, “I just don’t know what good this did.”

“I felt sad they felt the need to tear apart a historic place in Atlanta, but I also think we’re better than that,” she added. But, she said, “They felt this was the only way they could get their point across, and that’s sad.”

For those on the right, concern both about George Floyd’s death and the destructive protests that followed.

In her typical appearances on Fox News, Jeanine Pirro, a former Republican district attorney, reserves her highest dudgeon for castigating liberals and lamenting the demise of law and order.

But on Friday’s “Fox & Friends,” Ms. Pirro’s voice nearly broke as she described the agonizing final moments of George Floyd, the black man who died after a Minneapolis police officer ignored his pleas and pinned him to the ground during a routine stop.

“George Floyd was begging, saying he couldn’t breathe, saying please, please,” Ms. Pirro told viewers. “This man who put his knee on the neck of George Floyd does not deserve to be free in this country.”

Even right-wing stars like Rush Limbaugh hedged their assessments early on, as the officer’s lethal force drew more condemnation in some corners of the right than the ensuing riots and the burning of a police precinct. “I can’t find a way to justify it,’’ Mr. Limbaugh said of the officer’s actions.

The chilling circumstances of Mr. Floyd’s death — particularly the graphic, indisputable video of his arrest — have, at least for now, posed a political quandary among some conservative politicians, media stars and President Trump, whose usual instinct is to focus on blaming liberals for promoting lawlessness.

The ongoing protests in Minneapolis and around the country may still alter conservative views. On Fox News on Friday night, Tucker Carlson began his show with a graphic calling the Minnesota protesters “Criminal Mobs,” and wondered aloud why Republicans were not reacting more intensely against the violence in Minneapolis. Sean Hannity and Laura Ingraham condemned the demonstrators for, in Mr. Hannity’s words, “exploiting” Mr. Floyd’s death.

The officer who pinned Mr. Floyd was charged with murder, and his wife is seeking a divorce.

The intensifying protests came after the authorities announced that the officer who pinned George Floyd to the ground had been arrested and charged with murder on Friday, a development that activists and Mr. Floyd’s family had called for but also said did not go far enough.

The officer, Derek Chauvin, 44, was charged with third-degree murder and second-degree manslaughter, charges that come with a combined maximum sentence of 35 years.

An investigation into the three other officers who were present at the scene remains ongoing.

Mr. Floyd’s relatives have said that had wanted the more serious charge of first-degree murder.

Third-degree murder does not require an intent to kill, according to the Minnesota statute, only that the perpetrator caused someone’s death in a dangerous act “without regard for human life.” Charges of first- and second-degree murder require prosecutors to prove, in almost all cases, that the perpetrator made a decision to kill the victim.

A lawyer for Mr. Chauvin’s wife, Kellie, said that she was devastated by Mr. Floyd’s death and expressed sympathy for his family and those grieving his loss. The case has also led Ms. Chauvin to seek a divorce, the lawyer, Amanda Mason-Sekula, said in an interview on Friday night.

In a move that has since prompted protests in cities across the country, Mr. Chauvin pinned Mr. Floyd behind a police vehicle outside the store. For eight minutes and 46 seconds, according to a criminal complaint filed on Friday by the Hennepin County attorney, the police officer pressed his knee into Mr. Floyd’s neck in silence, staring toward the ground as his captive gasped repeatedly that he could not breathe.

Bystanders waved their cellphones, cursed and pleaded for help, and still, for two minutes and 53 seconds after Mr. Floyd had stopped protesting and became unresponsive, the officer continued to kneel.

The fatal encounter began just before 8 p.m. on Monday, when Mr. Floyd entered Cup Foods and a store clerk claimed that he had paid for cigarettes with a counterfeit $20 bill. In the minutes that followed, Mr. Floyd found himself on the ground, beneath the officer’s knee. He called, records say, for his mother. He said, “Please.”

One of the officers dismissed his pleas that he could not breathe.

“You are talking fine,” one officer said, according to the charging documents.

At least one officer was worried: That officer asked if they should roll Mr. Floyd over on his side.

“No, staying put where we got him,” Mr. Chauvin replied.

At 8:24 p.m., Mr. Floyd stopped moving.