Donald J. Trump rolled to victory in the Republican presidential primaries in Florida, Illinois and North Carolina on Tuesday, driving Senator Marco Rubio from the race and amassing a formidable delegate advantage that will be exceedingly difficult for any rival to overcome.

But with a victory in Ohio, his home state, Gov. John Kasich denied Mr. Trump one of the night’s biggest prizes and made it considerably harder for him to clinch the nomination outright before primary voting ends in June.

Senator Ted Cruz of Texas finished second in Illinois and North Carolina and was locked in a tight race with Mr. Trump in Missouri, ensuring that he, too, would earn a share of delegates.

Mr. Trump has faced mounting criticism from Republicans for the vitriolic tone of his candidacy, but he struck a defiant note Tuesday night, describing himself proudly as a candidate of the angry and disaffected. “There is great anger,” he said. “Believe me, there is great anger.”

Republicans opposed to Mr. Trump believe that Tuesday’s results may have increased their chances of denying him the nomination at the party’s convention in Cleveland. But they are left with a pair of deeply flawed alternatives: Mr. Cruz, who has the second-most delegates but is reviled by many party leaders, and Mr. Kasich, who has so far run the equivalent of a favorite-son campaign, winning only Ohio.

Mr. Kasich must now strain for a larger role in a Republican contest in which he has largely competed in obscurity. In his Tuesday night speech, he did not take on Mr. Trump by name, but said he would carry his own message of uplift “all the way to Cleveland.” 

Marco Rubio Suspends Campaign

“It’s been my intention to make you proud,” Mr. Kasich told a roaring crowd in Berea, Ohio, adding a favorite line: “I will not take the low road to the highest office in the land.”

Despite Mr. Kasich’s victory in Ohio, Mr. Cruz made another attempt on Tuesday to define the Republican race as the head-to-head contest he has long sought with Mr. Trump.

“After tonight, America has a clear choice going forward,” Mr. Cruz said once again, nodding to Mr. Rubio’s withdrawal. “Only one campaign has beaten Donald Trump over and over and over again.”

Mr. Trump remains the dominant figure in the race, however: His performance in Florida earned him 99 delegates and made a resounding statement about the appeal of his hard-edged populism in the country’s most sought-after swing state.

Mr. Rubio, addressing supporters in Miami, acknowledged that his campaign had been overwhelmed by an angry mood in the Republican electorate. In detached and clinical language, he said it had been impossible to repel the long-term political forces powering Mr. Trump.

“America’s in the middle of a real political storm — a real tsunami,” he said. “And we should have seen this coming.”

After congratulating Mr. Trump, Mr. Rubio essentially scolded him for the kind of campaign he has run.

“From a political standpoint, the easiest thing to have done in this campaign is to jump on all those anxieties,” Mr. Rubio warned in a valedictory address that at times sounded more like a policy seminar. “But that is not what’s best for America,” he added. “The politics of resentment against other people are not going to just leave us as a fractured party. They’re going to leave us as a fractured nation.”

Despite his triumph over Mr. Rubio, Mr. Trump was thwarted in his efforts to drive a second mainstream Republican rival from the race. Mr. Kasich’s victory in Ohio dealt Mr. Trump a stinging blow, preventing him from claiming the state’s 66 delegates and significantly increasing the chances that the Republican race will not be decided until the July convention.

It was the latest twist in an extraordinary campaign. Just three weeks ago, Republican leaders were complaining that Mr. Kasich, who until Tuesday had not won a single state, was ensuring Mr. Trump’s nomination by remaining in the race. Now, he has revived hopes throughout the party that Mr. Trump can be stopped.

Mr. Kasich’s victory in Ohio also shined a light on a nagging difficulty for Mr. Trump, and one of the Republican race’s most revealing divides: the class fault line. While Mr. Trump won among voters who earn less than $50,000 a year, Mr. Kasich overwhelmed him by more 30 percentage points among Ohioans who make more than $100,000.

Despite his Ohio win, Mr. Kasich was likely to emerge with fewer than 150 delegates over all — less than Mr. Rubio’s total. He alluded to heading west in “a covered wagon” and competing in California on June 7.

But in an important strategic signal, he also rolled out a list of longtime Republican insiders, some of them with experience in convention battles, who could be expected to press his case in Cleveland.

Mr. Kasich, after all, remains far behind Mr. Trump and Mr. Cruz in the delegate battle, and he has not come close to winning outside a small handful of states.

Mr. Trump’s performance in North Carolina was a reminder of his enduring strength in the South, but he was likely to take only a handful more delegates from the state than Mr. Cruz, who finished second, because North Carolina allocates its 72 delegates on a proportional basis.

Mr. Trump fared better in Illinois, where most delegates are allocated by congressional district. After making headlines for canceling a rally that nearly became a riot on Friday in Chicago, Mr. Trump easily won Cook County and appeared poised to take most of the delegates in the city. And while Mr. Cruz seemed likely to capture at least one congressional district downstate, Mr. Trump also showed strength across much of southern Illinois.

In a scenario that once would have been unthinkable for mainstream Republicans, they are now largely relying on Mr. Cruz, who made his name in the Senate vilifying party leaders, to slow Mr. Trump’s march to the 1,237 delegates he needs.

National Republican leaders had held out hope that Mr. Rubio could mount a comeback in Florida and challenge Mr. Trump for the nomination. But Mr. Rubio’s loss extinguished that possibility and left a multi-ballot convention fight this summer as mainstream Republicans’ last avenue to block Mr. Trump with a more palatable alternative.

With 358 delegates on the line, Tuesday’s voting opened a new phase in the Republican race, with states now permitted to award all their delegates to the victor. Even more significant, more than half of the states and territories have now voted, and the prospects of halting Mr. Trump’s campaign are waning.

Mr. Trump has proved an unexpectedly durable front-runner, with an unchallenged base of support among disaffected white voters who lack college degrees. But he has been unsuccessful in ending the nomination fight early, and each time he has seemed to gain a clear upper hand, he has given mainstream Republicans a spectacular reminder of why they have long resisted him.

Two days before his Super Tuesday victories on March 1, he startled the party by initially declining to disavow the endorsement of David Duke, the white supremacist and former Ku Klux Klan leader. And after Friday’s near-riot in Chicago, Mr. Trump made supportive-sounding commentsabout a backer who had punched a protester at a campaign rally in North Carolina. (On Tuesday night, supporters of Mr. Trump turned the tables in a way, interrupting the speeches of Mr. Rubio and Mr. Kasich.)

At the same time, an organized campaign to stop Mr. Trump has steadily gained force. Buoyed by an influx of money from major Republican donors, three advocacy groups have attacked him with television ads highlighting his business failures, his role in a defunct education company that is being sued for fraud, his extensive use of profanity and his history of misogynistic remarks.

Mr. Trump said Tuesday that he was baffled by his imperviousness to attack. Confronted with what he called “mostly false” criticisms, he said, “my numbers went up.”

“I don’t understand it,” he said Tuesday night, in a rare expression of disbelief at his own success. “Nobody understands it.”

Still, there were mounting signs that the primary battle has taken a toll on his candidacy. In all five of the states that held primaries, slightly less than half of Republican voters said they considered Mr. Trump honest and trustworthy, according to exit polls. And he has trailed Hillary Clinton in general election polls, often by wide margins.

Mr. Trump’s opponents have also been increasingly bold in confronting him, albeit belatedly. They have criticized his character and raised the prospect that they might be unwilling to cast a vote for him under any circumstances — even in a general election against Hillary Clinton.