President Biden on Friday dismissed concerns about his age, his mental acuity and polls showing him losing his re-election bid, saying in a prime-time interview that his sharpness is tested every day while he is “running the world.” He vowed to drop out only if “the Lord Almighty” tells him to.

During a 22-minute interview with ABC’s George Stephanopoulos, which aired unedited, Mr. Biden, 81, said there was no need for him to submit to neurological or cognitive testing. He said he simply did not believe the polls showing him losing. And asked how he would feel if former President Donald J. Trump were elected in November, he brushed off the question.

“I feel as long as I gave it my all and I did as good a job as I know I can do, that’s what this is about,” Mr. Biden said in an interview that was intended to assuage growing concerns about his age following last Thursday’s debate. But speaking in a hoarse voice and defiant throughout, there was little indication that the interview would do much to staunch the bleeding during the deepest crisis of this long political career.

Again and again, Mr. Biden told Mr. Stephanopoulos that voters should consider his accomplishments in office.

“Who’s going to be able to hold NATO together like me?” he said. “Who’s going to be able to be in a position where I’m able to keep the Pacific Basin in a position where, at least we’re checkmating China now? Who’s going to do that? Who has that reach?”

Mr. Biden repeatedly waved off “hypothetical” questions about whether he would step aside for another Democrat if people he respects say that he can’t win in the fall.

“Look, I mean, if the Lord Almighty came down and said, ‘Joe, get out of the race,’ I’d get out of the race, but the Lord Almighty’s not coming down,” Mr. Biden told Mr. Stephanopoulos. He dismissed concerns by Democratic lawmakers as overblown.

“Have you ever seen a group, a time when elected officials running for office aren’t a little worried? Have you ever seen that? I’ve not. Same thing happened in 2020,” he said, lowering his voice to mock officials who question his campaigning. “‘Oh, Biden, I don’t know what he’s going to do. He may bring me down.’”

Asked if he truly believed he was not trailing Mr. Trump in the race, he said that “all the pollsters I talk to say it’s a tossup — it’s a tossup.” And he said he was willing to take the risk that he is wrong about that.

“I don’t think anybody’s more qualified to be president or win this race than me,” he told Mr. Stephanopoulos.

The fact that the president was confronted with questions about his mental competency underscored the depth of crisis he is facing after the debate in Atlanta last week raised questions about his candidacy. A growing number of donors and several lawmakers have called for him to exit the race.

The president challenged that reality on Friday, insisting that “the vast majority are not where those folks are.” And he said no one around him has suggested that he needed to submit to an independent neurological examination.

“No. No one said I had to. They said I’m good,” he said. “Look, I have a cognitive test every single day. Every day I’ve had tests, everything I do. Not only in my campaign, but I’m running the world. And that sounds like hyperbole, but we are the essential nation in the world.”

Mr. Biden consented to the ABC interview — one of the few that he has given to news organizations during his presidency — and traveled to Madison, Wis., for a campaign rally in the hopes that strong performances could help rescue his teetering presidential campaign.

It was his first major interview since the debate, and he faced tougher questions than he did during a set of friendly interviews that aired Thursday with two Black talk radio hosts, during which he stumbled on his words and made a pair of verbal gaffes.

But it is far from clear that the interview or a routine rally, delivered with a teleprompter and seen by just a fraction of the millions who watched the debate, can begin to repair the political damage to his campaign, despite the fact that he largely avoided any major stumbles like the ones that shocked so many people during the debate last week.

In the interview, Mr. Biden struggled to explain away that debate performance, once again blaming it on a “bad cold” and appearing to suggest that he was caught flat footed by Mr. Trump’s barrage of lies.

“The whole way I prepared — nobody’s fault but mine, nobody’s fault but mine,” he said in a meandering answer. “I prepared for that I usually would do sitting down as I did come back with foreign leaders or the National Security Council, for explicit detail.”

He added: “The fact of the matter is that what I looked at is that he also lied 28 times. I couldn’t. I mean, the way that the debate ran, not — my fault, no one else’s fault. No one else’s fault.”

When Mr. Stephanopoulos noted that he seemed to struggle from the first minutes of the debate, Mr. Biden said: “Well, I just had a bad night.”

In one exchange that echoed Mr. Trump’s obsession with crowd size, Mr. Biden bragged about Friday’s rally where a several hundred people packed a small gym. He asked, “How many people you think can draw crowds like I drew tonight? You find many more enthusiastic than today? Huh?”

Mr. Stephanopoulos responded: “I don’t think you want to play the crowd game. Donald Trump can draw big crowds.”

The interview with Mr. Stephanopoulos was broadcast in full just hours after Mr. Biden vowed to stay in the race in front of the boisterous audience at the Madison, Wis., campaign rally, telling hundreds in the crowd that he will ignore calls for him to make way for another nominee.

“Guess what? They’re trying to push me out of the race,” he said. “Well, let me say this as clear as I can. I’m staying in the race!”

In his remarks on Friday, held inside a small, middle-school gym, he took on the issue of his age directly, saying he was not too old to create 15 million jobs, put the first Black woman on the Supreme Court or “beat big Pharma.”

“I’m in Wisconsin for one reason,” he said, “because we’re going to win.”

Mr. Biden’s actions in the days since the debate are taking place under an intense political microscope. Every word Mr. Biden uttered during the interview and rally is being viewed through the lens of the twin questions hanging over his campaign: At 81, is he too old? And can he still win?

For days, Mr. Biden’s team has said no, he’s not, and yes, he can.

But it took more than a week for the president to appear at the rally in Madison and in the interview with ABC’s George Stephanopoulos, letting anger fester for days as Democrats built momentum for the idea that he should quit the race.

A group of 168 business executives and donors issued a letter on Friday calling on him to step aside, including Paul Tagliabue, the former N.F.L. commissioner; John and Tom Florsheim, the shoe company brothers; and Christy Walton, a Walmart heir.

Representatives Seth Moulton, Democrat of Massachusetts and Mike Quigley, Democrat of Illinois, joined calls by two other House Democrats for Mr. Biden to end his re-election bid. Mr. Moulton told a Boston radio station on Thursday that he should “follow in one of our founding father, George Washington’s, footsteps and step aside to let new leaders rise up.”

Senator Mark Warner, Democrat of Virginia, is working to convene Democratic senators next week to discuss a path forward, while Representative Hakeem Jeffries, Democrat of New York and the minority leader, has scheduled a virtual meeting on Sunday with senior House Democrats to discuss President Biden’s candidacy.

Throughout the day on Friday, Mr. Biden remained defiant and even testy.

In a brief exchange with reporters after sitting for the ABC interview, he accused the news media of having been “been wrong about everything” in predicting the outcome of elections. And he dismissed Mr. Warner as “the only one” in the Senate talking about encouraging him to drop out of the race.

“I’m completely ruling that out,” he told the reporters as he boarded Air Force One at Dane County Regional Airport, adding that he is “committing now, absolutely” to another debate against Mr. Trump. A second debate is scheduled for September.

Asked about a succession plan, he said: “By the way, we do have succession plans. But what do I need a succession plan for now?”