After video footage emerged which showed the Republican nominee making crude comments about women, he told reporters: “I’d never withdraw. I’ve never withdrawn in my life.”
He added later there was “zero chance I’ll quit” in the battle with Democratic rival Hillary Clinton.
The comments came as a growing number of Republican figures urged Mr Trump to leave the race for the White House after a video surfaced which captured him making vulgar and sexually charged comments about women.
Mr Trump said he has also been getting calls of support after the footage dating from 2005 emerged. His wife Melania issued a statement saying the comments did “not represent the man that I know”.
However, his running mate, the Republican vice presidential nominee Mike Pence, said he could not condone or defend Mr Trump’s comments about women.
Mr Pence said: “As a husband and a father, I was offended by the words and actions described by Donald Trump in the 11-year-old video released yesterday.”
“I am grateful that he has expressed remorse and apologised to the American people.”
Mrs Clinton called Mr Trump’s remarks “horrific”. She said in a Twitter message: “We cannot allow this man to become president.”
Republican leaders from Utah to Alabama called on Mr Trump to leave the presidential race as the party struggled to grapple with the crisis.
However, US House Speaker Paul Ryan and various other high-profile Republicans refused to abandon their nominee, who has long faced criticism from within his own party, but never to this degree.
In a videotaped apology, Mr Trump declared “I was wrong and I apologise”.
He also defiantly dismissed the revelations as “nothing more than a distraction” from a decade ago.
“I’ve said some foolish things,” Mr Trump said in the video. But there’s a big difference between the words and actions of other people. Bill Clinton has actually abused women and Hillary has bullied, attacked, shamed and intimidated his victims.”
Republican strategist Terry Sullivan, who previously led Marco Rubio’s presidential campaign, said of Mr Trump’s White House battle: “It’s over. The only good news is that in 30 days Trump will be back to being just a former reality TV star like the Kardashians, and Republican candidates across America will no longer be asked to respond to his stupid remarks.”