"I'd never withdraw. I've never withdrawn in my life," he told the Washington Post in a phone interview Saturday morning.
"No, I'm not quitting this race. I have tremendous support," he added.
Sen. Kelly Ayotte and other Republicans have abandoned the candidate, begging him to make way for his running mate Mike Pence to lead the ticket.
But in an interview with the New York Times, also on Saturday, Trump claimed to have not known about anyone calling for him to withdraw.
“I haven’t heard from anyone saying I should drop out, and that would never happen, never happen,” Trump said. “That’s not the kind of person I am. I am in this until the end.”
Trump tweeted Saturday afternoon that he would never let his supporters down.
Trump agreed that the video, first published by the Post, is now dominating the presidential race, but maintained that he has gone through similar difficulties.
“I’ve been here before, I’ll tell ya, in life,” he said. “I understand life and how you make it through. You go through things. I’ve been through many. It’s called life. And it’s always interesting.”
Donald Trump issued a rare videotaped apology early Saturday after a recording surfaced hours earlier of him bragging about trying to sleep with a married woman and making shockingly vulgar comments about women.
But the Republican presidential nominee struck a defiant tone in his response, saying the release of the 2005 tape was "nothing more than a distraction from the important issues we're facing today" that were not nearly as appalling as former President Bill Clinton's actions.
"Bill Clinton has actually abused women, and Hillary has bullied, attacked, shamed and intimidated his victims," Trump said.
The candidate said that despite the crude language he used while talking with Billy Bush of Access Hollywood, he is a "changed man" and pledged "to be a better man tomorrow, and I will never, ever let you down."
"I've never said I'm a perfect person, nor pretended to be someone I'm not," he said. "I've said and done things I regret and the words released today on this more than a decade old video are one of them. Anyone who knows me knows these words don't reflect who I am. I said it, I was wrong, and I apologize."
The message came just hours after the Washington Post published a video of Trump in which he was caught on a hot mic describing how he tried to have sex with a married woman.
“I did try and fuck her," he said. "She was married.”
He later on the tape that if you're a "star" you can "do anything," including grab women "by the pussy."
The comments were made in 2005, months after Trump married his third wife, Melania.
The comments in the Post video prompted a cascade of Republican criticism, with many in the candidate's party demanding an apology and several calling for him to drop out. The criticism was particularly strong from Utah, where Gov. Gary Herbert and Congressman Jason Chaffetz became the first Republicans to say they had changed their minds and would no longer vote for their party's nominee.