Mr Biden, 80, twice told a campaign reception in Manhattan on Wednesday about the 2017 Charlottesville riots during a 20-minute talk.
At the event, hosted by billionaire real estate heiress Amy Goldman Fowler, Mr Biden told the audience he had “no intention of running” for the White House following Barack Obama’s two administrations.
But then, he said, “along came, in August of 2017, Charlottesville, Virginia.”
“You remember those folks walking out of the fields literally carrying torches, with Nazi swastikas, holding them forward, singing the same vicious, antisemitic bile — the same exact bile — bile that was sung in — in Germany in the early ‘30s. And a young woman was killed. A young woman was killed”, he added.
According to the official White House transcript, Mr Biden then recounted that Donald Trump was asked, following the rally, “What do you think would happen?”
“He was the sitting president”, Mr Biden said. “And he said, ‘I thought there were some very fine people on both sides.’ And I mean this sincerely, from the bottom of my heart, that’s when I decided I — I was going to run again.”
Mr Biden went on to describe how his grandchildren had urged him to challenge Mr Trump despite the anticipated blowback.
But the US President then began to talk about Charlottesville again.
He said: “You may remember that, you know, those folks from Charlottesville, as they came out of the fields and carrying those swastikas, and remember the ones with the torches and the Ku — accompanied by the Ku Klux Klan.
“And in addition to that, they had — there were white supremacists. Anyway, they were making the big case about how terrible this was. And a young woman was killed in the process.
“And my predecessor, as I said, was asked what he thought. He said, ‘There are some very fine people on both sides.’ Well, that kept ringing in my head.
“And so, I couldn’t, quite frankly, remain silent any longer,” Mr Biden added. “So, I decided I would run. And it became — I ran because I thought everything this country stood for was up for grabs for the first time in my career.”
During the speech, Mr Biden also referred to his age, saying: “as you might have noticed, a lot of people seem a little focused on my age.
“I get it. And believe me, I know better than anyone.”
He then went on to say “with age comes wisdom”.
A poll published by the Wall Street Journal earlier this month found 73 per cent of Americans, including two thirds of Democrats, think Mr Biden is too old to run again in next November’s presidential election.