Donald Trump said he will carry on as Republican candidate for president – despite numerous calls from Republicans for him to drop out – following the revelation of vulgar remarks he made about women in 2005.

“I’d never withdraw. I’ve never withdrawn in my life,” he told the Washington Post. “No, I’m not quitting this race. I have tremendous support.

“People are calling and saying: ‘Don’t even think about doing anything else but running’,” Mr Trump said when asked about Republican defection. “You have to see what’s going on. The real story is that people have no idea the support. I don’t know how that’s going to boil down but people have no idea the support.”

“They’re not going to make me quit, and they can’t make me quit,” Mr Trump said, responding to calls for him to step down. “The Republicans, you’ve got to remember, have been running for a long time. The reason they don’t win is because they don’t stick together.”

He claimed that running against Hillary Clinton gives him enough default support from the Republican Party, all but suggesting he still has a strong footing in the race.

“It's because she’s so bad. She’s so flawed as a candidate,” he said. “Running against her, I can’t say it be the same if I ran against someone else, but running against her makes it a lot easier, that's for sure.”

In a later interview with the Wall Street Journal, Mr Trump said that his wife, Melania, and daughter, Ivanka, “understand” his situation and “they’re very loyal”.

Melania Trump issued a statement later Saturday afternoon, condemning the remarks, while also defending Mr Trump.

"The words my husband used are unacceptable and offensive to me," she said. "This does not represent the man that I know. He has the heart and mind of a leader. I hope people will accept his apology, as I have, and focus on the important issues facing our nation and the world."

Donald Trump apologises for 'grab them by the p****' remarks, says Bill Clinton has done 'far worse'

Despite the apology he issued early Saturday morning, he shifted the focus to his critics.

“Go behind closed doors of the holier-than-thou politicians and pundits and see what they’re saying. I look like a baby,” he said.

Mr Trump’s remarks came a day after a 2005 video emerged showing the Republican candidate bragging in lewd terms about sexually assaulting women with impunity.

“I moved on her actually, she was down in Palm Beach and I failed. I’ll admit it. I did try to f**k her, she was married ... and I moved on her very heavily,” Mr Trump is heard telling then-Access Hollywood host Billy Bush.

Republican leaders backing away from Donald Trump after lewd remarks Show all 12 1 /12 Republican leaders backing away from Donald Trump after lewd remarks Republican leaders backing away from Donald Trump after lewd remarks Former governor of California Arnold Schwarzenegger Schwarzenegger said he will not vote for Trump Getty Republican leaders backing away from Donald Trump after lewd remarks Senate Majority Leader Mitch McConnell Mr McConnell condemned the remarks, but has not unendorsed him Getty Republican leaders backing away from Donald Trump after lewd remarks House Speaker Paul Ryan Mr Ryan disinvited Mr Trump from a Wisconsin campaign event following the remarks Getty Republican leaders backing away from Donald Trump after lewd remarks Former Sec of State Condoleezza Rice Ms Rice called for Trump to withdraw his candidacy Getty Republican leaders backing away from Donald Trump after lewd remarks Nebraska Sen Deb Fischer Ms Fischer called on Trump to step aside and give nomination to Mike Pence Getty Republican leaders backing away from Donald Trump after lewd remarks Texas Sen Ted Cruz Mr Cruz denounced the remarks, but still endorses Trump Getty Republican leaders backing away from Donald Trump after lewd remarks Former Republican candidate Carly Fiorina Ms Fiorina wants Mike Pence to take the nomination Getty Republican leaders backing away from Donald Trump after lewd remarks Conservative radio host Hugh Hewitt Hewitt called for Trump to step down Getty Republican leaders backing away from Donald Trump after lewd remarks Utah Representative Mia Love Ms Love said she cannot vote for Trump after hearing his remarks Getty Republican leaders backing away from Donald Trump after lewd remarks Former New York Gov George Pataki Mr Pataki said Mr Trump should step down Getty Republican leaders backing away from Donald Trump after lewd remarks New Hampshire Sen Kelly Ayotte Ms Ayotte will write in Mike Pence when she votes Getty Republican leaders backing away from Donald Trump after lewd remarks Arizona Sen John McCain Mr McCain will choose Mike Pence on Election Day Getty

After Mr Trump makes this comment, he and Mr Bush appear to notice Days of Our Lives actor Arianne Zucker, who was there to escort them to the set of the soap opera.

“Your girl’s hot as sh**, in the purple,” Mr Bush, now a host on NBC's Today show, said.

“I’ve gotta use some Tic-Tacs, just in case I start kissing her,” Mr Trump says in the video. “You know I’m automatically attracted to beautiful – I just start kissing them. It’s like a magnet. Just kiss. I don’t even wait.”

“And when you’re a star they let you do it,” he adds. “You can do anything… Grab ’em by the p****.”

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The video caused a massive backlash from conservative supporters in subsequent hours.

Mr Trump’s running mate, Indiana Governor Mike Pence, released a statement Saturday condemning the remarks, shortly after House Speaker Paul Ryan disinvited the Republican nominee from a GOP campaign event in Wisconsin scheduled for Saturday.

“As a husband and father, I was offended by the words and actions described by Donald Trump in the 11-year-old video released yesterday,” Mr Pence said, after cancelling his appearance in Mr Ryan's home state. “I do not condone his remarks and cannot defend them. I am grateful that he has expressed remorse and apologised to the American people.”

Mr Ryan issued a sharp rebuke of Mr Trump's remarks in the wake of the video's release.

“I am sickened by what I heard today. Women are to be championed and revered, not objectified,“ Mr Ryan said on Friday. “I hope Mr Trump treats this situation with the seriousness it deserves and works to demonstrate to the country that he has greater respect for women than this clip suggests.

“In the meantime, he is no longer attending tomorrow’s event in Wisconsin.”

15-year-old girl: I see with my own eyes the damage Donald Trump does when he talks about women

Responding to Mr Trump’s comments, Hillary Clinton wrote on Twitter: “This is horrific. We cannot allow this man to become president.”

Senate majority leader Mitch McConnell called on Mr Trump to take responsibility for his comments.

“As the father of three daughters, I strongly believe that Trump needs to apologise directly to women and girls everywhere, and take full responsibility for the utter lack of respect for women shown in his comments on that tape,” he said.

New Hampshire Senator Kelly Ayotte released a statement on Saturday morning retracting her endorsement for the New York business mogul.

“I wanted to be able to support my party’s nominee, chosen by the people, because I feel strongly that we need a change in direction for our country,” she said. “However, I'm a mom and an American first, and I cannot and will not support a candidate for president who brags about degrading and assaulting women. I will not be voting for Donald Trump or Hillary Clinton and instead will be writing in Governor Pence for president on election day.”

Ms Ayotte’s sentiment echoed a simple statement made by Utah congressman Jason Chaffetz: “I wish that Mike Pence was at the top of the ticket.”

US election 2016: the race for the White House in pictures Show all 12 1 /12 US election 2016: the race for the White House in pictures US election 2016: the race for the White House in pictures Republican U.S. presidential nominee Donald Trump shakes hands with Democratic U.S. presidential nominee Hillary Clinton at the conclusion of their first presidential debate at Hofstra University in Hempstead, New York Reuters US election 2016: the race for the White House in pictures President Barack Obama embraces Democratic nominee Hillary Clinton on stage at the party's convention in Philadelphia US election 2016: the race for the White House in pictures Donald Trump's wife Melania delivered a speech at the GOP convention in Cleveland that was later found to have been cribbed in part from Michelle Obama's 2008 convention address AP US election 2016: the race for the White House in pictures Hillary Clinton talks to reporters aboard her new campaign plane on Labour Day, 5 September, her first 'press conference' since 2015 (Getty Images) US election 2016: the race for the White House in pictures Donald Trump held a joint press conference with Mexican leader Enrique Pena Nieto in Mexico City in August, hours before reiterating his harsh immigration plans at a campaign rally in Arizona Reuters US election 2016: the race for the White House in pictures Bernie Sanders officially endorsed Hillary Clinton, saying his progressive vision for ‘a transformed America’ would be ’best served by the defeat of Donald Trump’ Reuters US election 2016: the race for the White House in pictures Khizr and Gazala Khan appeared at the DNC to slam Trump for his stance on Muslim immigration, citing the case of their son Humayun Khan, who was killed in combat while serving as a Captain in the US Army in Iraq US election 2016: the race for the White House in pictures Libertarian candidate Gary Johnson is doing better in polls than any third party candidate since Ross Perot, 20 years ago Nicholas Kamm/AFP/Getty US election 2016: the race for the White House in pictures Green Party candidate Jill Stein (centre) marches with supporters in Colorado AP US election 2016: the race for the White House in pictures Hillary Clinton and Virginia Senator Tim Kaine at a rally in Kaine's home state in July, days before Ms Clinton tapped him to be her running mate Getty US election 2016: the race for the White House in pictures Trump on the campaign trail with his vice presidential pick, Indiana governor Mike Pence AP US election 2016: the race for the White House in pictures Former Ukip leader Nigel Farage appears at a Trump rally in Mississippi in August, where he told the crowd that he 'wouldn't vote for Hillary Clinton if you paid me'.

Conservative radio host Hugh Hewitt also called for Mr Trump to withdraw from the race.

“For the benefit of the country, the party and his family, and for his own good, [Donald Trump] should withdraw. More and worse [opposition] coming,” he tweeted.

Former Hewlett-Packard CEO and Republican candidate Carly Fiorina – who herself was the target of misogynistic insults from Mr Trump last year – rebuked the nominee in a statement on Facebook.

“Donald Trump does not represent me or my party,” Ms Fiorina said. “I understand the responsibility of Republicans to support their nominee. Our nominee has weighty responsibilities as well. Donald Trump has manifestly failed in these responsibilities.”

Mr Trump initially wrote off the controversy as nothing more than locker room banter among boys, but after haemorrhaging supporters for hours, he issued a video apology at 12.30am Saturday.

Nonetheless, Mr Trump denies the significance of the Republican exodus from his corner of the ring, claiming in his interview with the Washington Post that he has received “thousands and thousands” of letters encouraging him to carry on.

In a statement released on Saturday night his wife Melania Trump, who was pregnant at the time of the video, said: “The words my husband used are unacceptable and offensive to me. This does not represent the man that I know. He has the heart and mind of a leader. I hope people will accept his apology, as I have, and focus on the important issues facing our nation and the world.”

Ahead of the Sunday night debate with Ms Clinton, Mr Trump indicated that he may give a speech from New York to encourage the supporters who still remain by his side. Among public figures, however, that number remains very small.

Fox News’s Sean Hannity defended Mr Trump by invoking biblical trivia.

“King David had 500 concubines for crying out loud,” he said.

While conservative commentator Dinesh D’Souza accused the press of conspiring to send Ms Clinton to the White House.