President Biden on Monday dared his critics to “challenge me at the convention” if they want him out of the presidential race, refusing to step aside in a defiant letter to Democratic members of Congress and in fiery remarks on MSNBC’s “Morning Joe” program.

Declaring himself “frustrated by the elites” who have called for his exit from the race, Mr. Biden engaged in an offensive blitz aimed at saving his candidacy. But it also did little to mollify restless Democratic lawmakers and laid bare the fractures in his party over whether his position as its standard-bearer will help or hurt its fortunes this fall.

In effect, Mr. Biden decided to engage in a no-holds-barred fight with his allies for the world to see. He was at turns defiant, furious, indignant, exasperated and dismissive. He insisted that he would not withdraw from the race but accused those who have suggested he step aside of being routinely wrong about politics.

The president used the friendly venue of the morning news show — hosted by Joe Scarborough, a longtime supporter who recently has been critical — to respond to demands that he demonstrate the kind of vigor that was missing from his listless and at times incoherent debate performance on June 27.

Mr. Biden raised his voice repeatedly during the brief phone interview, including after Mika Brzezinski, the co-host of the show and Mr. Scarborough’s wife, asked him whether he had undergone neurological exams after the debate. Sounding exasperated and angry, Mr. Biden rejected assertions that his stamina and mental abilities have not been tested in a real way.

“It drives me nuts, people talking about this,” he said.

In the days after the debate with former President Donald J. Trump, Mr. Biden tried to be conciliatory. He admitted that he had “a bad night” and tried to explain that his performance was the result of a cold and jet lag.