Over the course of his long career, President Biden has overcome personal tragedy and political odds, and he has used his resilience to power his ambition. But now that he is in the fight of his political life, his irrepressible pursuit of the comeback risks looking like blind defiance in the face of a rising tide.

“You’ve been wrong about everything so far,” Mr. Biden told a group of reporters who asked him on Friday why he still felt he was the best person to defeat former President Donald J. Trump, after a dismal debate performance in Atlanta plunged his campaign into crisis.

“You were wrong about 2020. You were wrong about 2022. We were going to get wiped out — remember the red wave,” he said, referring to an expected wave of Republican gains that never materialized in the midterm elections. Instead, Democrats did far better than expected, a decisive factor in Mr. Biden’s decision to run for a second term.

He took that spirit with him into a 22-minute ABC interview with George Stephanopoulos on Friday, in which he was asked about his approval rating of 36 percent.

“Well, I don’t believe that’s my approval rating,” Mr. Biden said. “That’s not what our polls show.”

In an effort to turn out his supporters, the president is scheduled to attend several campaign events in Pennsylvania with the first lady on Sunday.

The comeback-kid mythology that Mr. Biden has built over a half century in politics is colliding with a new reality, where he is not being held up as a fighter who overcomes obstacles but is accused by his critics of putting his own ego ahead of the country. His denial of polls, voter concerns and growing calls among Democrats for him to leave the race have left some in his party with a quandary about how aggressively to try to persuade him to change course, with time running short to make a change.

For Mr. Biden, the phrase “when you get knocked down, you get back up” is not simply a campaign ad. It is a key to understanding how he views himself. He sees himself as the scrappy, average-Joe politician who has been constantly overlooked, underestimated and counted out. To him, the naysayers are as necessary to his story as the supporters.

And he does have supporters who say they still believe in him, including Gov. Gavin Newsom, a Democrat of California who visited a Biden campaign office in Pittsburgh on Friday to boost morale among volunteers in a crucial state. But even Mr. Newsom, who has been full-throated with his support since the moment Mr. Biden exited the debate stage 10 days ago, was clear about the stakes: “This is a really important week and weekend. None of us are naïve about the pivot after that debate.”

Campaign officials said before the debate that Mr. Biden’s performance, good or bad, would not cause polls to crater.

“This was always going to be a close race,” said Kevin Munoz, a campaign spokesman, “and the dynamics at play are the ones we’ve long anticipated: Voters continue to be deeply concerned by Donald Trump and his harmful agenda, and the more we engage and reach out to voters, the more they support President Biden.”

On Saturday, Mr. Biden joined a call with a group of senior campaign surrogates and spent an hour and 15 minutes soliciting their feedback about the past few days. Senator Chris Coons of Delaware, a longtime Biden ally, said everyone on the call encouraged him to stay in the race but also relayed that supporters had a range of concerns about his ability to keep the job for another four years.

“He asked us to share, ‘Who have you heard from, who is critical or has a concern or isn’t persuaded?’” Mr. Coons said. Mr. Biden told the group that he understood the concerns, and wanted to do more interviews and unscripted appearances in the future.

Kate Bedingfield, who served as White House communications director until 2023, said Mr. Biden would still need to prove himself in the days ahead.

“I think this is a very competitive race but I also think that there’s no question that he has a lot of work to do here,” Ms. Bedingfield said. “This is a moment where he really needs to show people that he is all in it for the reasons that I know he’s in it, which is to protect democracy and defeat Donald Trump and push back on the threat that he poses to our country.”

The ABC interview was supposed to be one opportunity for him to do that. “He didn’t succeed,” David Axelrod, a veteran strategist and alumnus of the Obama administration, wrote in an opinion essay on Saturday. People close to Mr. Biden felt differently, that he’d done enough to continue fighting an establishment that shows signs of turning against him.

On Saturday, Mr. Biden huddled with family at his home in Wilmington, Del., and attended church with his sister, Valerie Biden Owens. His family has urged him to stay in the fight more often than not. Last week, the Bidens gathered at Camp David, where again, their message was the same: Stay in. The family is still urging Mr. Biden to stay the course, according to people familiar with their thinking.

Jill Biden, the first lady, has traveled the country to campaign in recent days. Hunter Biden, Mr. Biden’s son, is pushing for his father to stay in the race.

According to a half dozen people familiar with the Biden family dynamics, Mr. Biden may be fielding advice from allies and placing calls to Democrats on Capitol Hill in an effort to keep the dam from breaking against him, but they say his decisions in the coming days and weeks will be heavily influenced by his family.

Democrats have expressed distress in recent days that Mr. Biden’s family could have so much control over his political future. People who know the family, however, say it has always been this way.

Mr. Biden recounted in his memoir, “Promises to Keep,” that when he was considering dropping out of the presidential primary race amid a plagiarism scandal in 1987, his two sons, Beau and Hunter, caught him in the hallway at home. The pair pleaded with him to stay in the race. They feared how much he might change if he was not running to prove himself and to show the world who he was.

“The only thing that’s important is your honor,” Hunter, who was in his teens, told his father at the time. “That’s what you’ve always taught us. Your honor.”

“You’ll change, Dad,” warned his son, Beau, who died in 2015. “You’ll never be the same.”

Mr. Biden eventually left that race, but these are different circumstances. Beau Biden is gone, and it is a loss that has both gutted Mr. Biden and guided his presidency. Hunter Biden has been convicted of three felony gun charges, and his ongoing legal problems are said to be a major personal weight on the president.

On top of that, Mr. Biden is now an 81-year-old incumbent. And Democrats closest to him still privately fear what would happen if anyone but him is on the ticket against Mr. Trump.

Mr. Biden has been adamant that his debate showing was a one-off, and says he wants to debate Mr. Trump again. He told Mr. Stephanopoulos to watch as American allies descend on Washington next week for a NATO meeting, where he will work to shore up relations and deliver a news conference.

“Come listen,” Mr. Biden beckoned his interviewer, forecasting the widespread support that he has always believed has been his due. “See what they say.”