WASHINGTON — With a nation on edge, ravaged by disease, hammered by economic collapse, divided over lockdowns and even face masks and now convulsed once again by race, President Trump’s first instinct has been to look for someone to fight.

Over the last week, America reeled from 100,000 pandemic deaths, 40 million people out of work and cities in flames over a brutal police killing of a subdued black man. But Mr. Trump was on the attack against China, the World Health Organization, Big Tech, former President Barack Obama, a cable television host and the mayor of a riot-torn city.

While other presidents seek to cool the situation in tinderbox moments like this, Mr. Trump plays with matches. He roars into any melee he finds, encouraging street uprisings against public health measures advanced by his own government, hurling made-up murder charges against a critic, accusing his predecessor of unspecified crimes, vowing to crack down on a social media company that angered him and then seemingly threatening to meet violence with violence in Minneapolis.

As several cities erupted in street protests after the killing of George Floyd, some of them resulting in clashes with the police, Mr. Trump made no appeal for calm. Instead in a series of tweets and comments to reporters on Saturday, he blamed the unrest on Democrats, called on “Liberal Governors and Mayors” to get “MUCH tougher” on the crowds, threatened to intervene with “the unlimited power of our Military” and even suggested his own supporters mount a counterdemonstration.

The turmoil came right to Mr. Trump’s doorstep for the second night in a row on Saturday as hundreds of people protesting Mr. Floyd’s death and the president’s response surged in streets near the White House. While most were peaceful, chanting “black lives matter” and “no peace, no justice,” some spray painted scatological advice for Mr. Trump, ignited small fires, set off firecrackers and threw bricks, bottles and fruit at Secret Service and United States Park Police officers, who responded with pepper spray.

The police cordoned off several blocks around the Executive Mansion as a phalanx of camouflage-wearing National Guard troops marched across nearby Lafayette Square. A man strode through the streets yelling, “Time for a revolution!” The image of the White House surrounded by police in helmets and riot gear behind plastic shields fueled the sense of a nation torn apart.

Mr. Trump praised the Secret Service for being “very cool” and “very professional” but assailed the Democratic mayor of Washington for not providing city police officers to help on Friday night, which she denied. While governors and mayors have urged restraint, Mr. Trump seemed more intent on taunting the protesters, bragging about the violence that would have met them had they tried to get onto White House grounds.

“Big crowd, professionally organized, but nobody came close to breaching the fence,” the president wrote on Twitter on Saturday morning. “If they had they would have been greeted with the most vicious dogs, and most ominous weapons, I have ever seen. That’s when people would have been really badly hurt, at least. Many Secret Service agents just waiting for action.”

His suggestion that his own supporters should come to the White House on Saturday foreshadowed the possibility of a clash outside his own doors. “Tonight, I understand, is MAGA NIGHT AT THE WHITE HOUSE???” he wrote on Twitter, using the acronym for his first campaign slogan, “Make America Great Again.”

Asked about the tweet later, he denied encouraging violence by his supporters. “They love African-American people,” he said. “They love black people. MAGA loves the black people.” By evening, however, Mr. Trump’s supporters were not in evidence among the crowds at the White House.

Mayor Muriel E. Bowser of Washington responded sharply on Saturday morning, saying her police department will protect anyone in Washington, including the president, and by Saturday evening her officers were out in force around the White House.

But she called the president a source of division. “While he hides behind his fence afraid/alone, I stand w/ people peacefully exercising their First Amendment Right after the murder of #GeorgeFloyd & hundreds of years of institutional racism,” she wrote. “There are no vicious dogs & ominous weapons. There is just a scared man. Afraid/alone …”

After his morning barrage, Mr. Trump tried to recalibrate later in the day, devoting the opening of a speech at the Kennedy Space Center following the SpaceX rocket launch to the unrest in the streets and clearly trying to temper his bellicose tone.

“I understand the pain that people are feeling,” he said. “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 peace. The memory of George Floyd is being dishonored by rioters, looters and anarchists.”

The days of discord have put the president’s leadership style on vivid display. From the start of his ascension to power, Mr. Trump has presented himself as someone who seeks conflict, not conciliation, a fighter, not a peacemaker. That appeals to a substantial portion of the public that sees in him a president willing to take on an entrenched and entitled establishment.

But the confluence of perilous health, economic and now racial crises has tested his approach and left him struggling to find his footing just months before an election in which polls currently show him behind.

“The president seems more out-of-touch and detached from the difficult reality the country is living than ever before,” said Carlos Curbelo, a former Republican congressman from Florida who has been critical of Mr. Trump. “At a moment when America desperately needs healing, the president is focused on petty personal battles with his perceived adversaries.”

Such a moment would challenge any president, of course. It has been a year of national trauma that started out feeling like another 1998 with impeachment, then another 1918 with a killer pandemic combined with another 1929 given the shattering economic fallout. Now add to that another 1968, a year of deep social unrest.

It is fair to say that 2020 has turned out to be a year that has frayed the fabric of American society with an accumulation of anguish that has whipsawed the country and its people. But in some ways, Mr. Trump has become a totem for the nation’s polarization rather than a mender of it.

“I am daily thinking about why and how a society unravels and what we can do to stop the process,” said Timothy Naftali, a presidential historian at New York University. “The calamity these days is about more than Trump. He is just the malicious con man who lives to exploit our vulnerabilities.”

As the nation has confronted a coronavirus pandemic at the same time as the greatest economic catastrophe since the Great Depression, whatever unified resolve that existed at the beginning of the twin crises quickly evaporated into yet another cultural clash. And the president has made everything into just another partisan dispute rather than a source of consensus, from when and how to reopen to whether to wear a mask in public.

Mr. Trump led no national mourning as the death toll from the coronavirus passed 100,000 beyond lowering the flags at the White House, posting a single tweet and offering a passing comment on camera only when asked about it. Rather than seek agreement on the best and safest way to restore daily life, he threatened to “override” governors who prevented places of worship from resuming crowded services.

“Crisis leadership demands much more from the White House than irresponsible threats on social media,” said Meena Bose, director of the Peter S. Kalikow Center for the Study of the American Presidency at Hofstra University.

Mr. Trump’s initial response to the rioting in Minneapolis, where a police officer has been charged with murder after kneeling on Mr. Floyd’s neck for nearly nine minutes as he cried out that he could not breathe, underscored the president’s most instinctive response to national challenges. Threatening to send in troops, he wrote early Friday morning that “when the looting starts, the shooting starts.”

Only after a cascade of criticism did he try to walk it back, posting a new tweet 13 hours later, suggesting that all he had meant was that “looting leads to shooting” by people in the street.

“I don’t want this to happen, and that’s what the expression put out last night means,” he said, a reformulation that convinced few if any of his critics.

Even some of Mr. Trump’s usual allies were distressed at the original shooting tweet. Geraldo Rivera, the television and radio host who often spends time with Mr. Trump at the president’s Mar-a-Lago club in Florida, decried “the recklessness” of that message and called on the president “to self-censor himself.”

“Come on, what is this, sixth grade?” Mr. Rivera said on Fox News. “You don’t put gasoline on the fire. That’s not calming anybody.” He added: “All he does is diminish himself.”

But many of the president’s defenders rejected the idea that he had mishandled the crises, pressing the argument that Democrats and the news media were to blame for the turmoil in the streets, which spread from Minneapolis to New York, Atlanta, Washington, Louisville, Portland and other cities.

“Keep track of cities where hundreds of millions of dollars in property damage and serious injuries and death will take place,” Rudolph W. Giuliani, the former New York mayor who has served as Mr. Trump’s personal lawyer, wrote on Twitter on Friday night. “All Democrat dominated cities with criminal friendly policies. This is the future if you elect Democrats.”

Bernard B. Kerik, the former New York City police commissioner who was pardoned by Mr. Trump for tax fraud earlier this year, amplified the point on Twitter. “It should be no surprise that every one of these cities that the anarchist have taken over, are the same cities run by leftist Democrats with the highest violence, murder and poverty rates,” he wrote on Twitter. “They can’t handle their cities normally, so how are they going to deal with this?”

Mr. Trump, who this past week retweeted a video of a supporter saying that “the only good Democrat is a dead Democrat” (though the supporter insisted he meant that in a political sense), picked up the theme on Saturday.

With crowds visible from his upstairs windows, Mr. Trump reached for his phone and again assailed the “Democrat Mayor” of Minneapolis for not responding more vigorously and called on New York to unleash its police against crowds. “Let New York’s Finest be New York’s Finest,” he wrote. “There is nobody better, but they must be allowed to do their job!”