He told campaign donors Trump and his allies are out to “destroy" democracy.
"We've got to get it done, not because of me," Mr Biden told a fundraiser near Boston. "If Trump wasn't running I'm not sure I'd be running. We cannot let him win."
His forceful rhetoric came after Mr Trump, the front runner for the Republican party nomination, also called his successor in the White House the "destroyer of American democracy."
On Tuesday, Trump was asked by Fox News anchor Sean Hannity to promise he "would never abuse power as retribution against anybody."
"Except for day one," he responded. "I want to close the border and I want to drill, drill, drill. After that I'm not a dictator."
The Biden campaign seized on the comments with an email that read, "Donald Trump: Day One Dictator."
Asked by reporters whether he would be running if Trump was not, Mr Biden said: “I expect so, but look, he is running and I have to run."
He said he would not drop out of the race if his rival did.
“American democracy, I give you my word as a Biden, is at stake," the president said at the first of three campaign fundraisers.
“He didn't even show up at my inauguration. I can't say I was disappointed, but he didn't even show up.
"We're always going to defend, protect and fight for democracy. That's why I'm running."
Mr Biden will attend a fundraiser on Wednesday near the White House and another on Monday in Philadelphia.
On Friday, he will head to Los Angeles for a big-dollar event that will be his first since the writers’ strike, which ended in September.
The event is slated to be at the home of Michael Smith, a celebrity interior designer who decorated the White House for President Barack Obama.
Film director Steven Spielberg and his wife Kate Capshaw are among the hosts of the event.