New York (CNN) Donald Trump on Saturday vowed to "never" drop out of the presidential race as a growing chorus of Republicans urged him to do exactly that after sexually aggressive remarks he made in 2005 surfaced a day earlier.

The Republican nominee, eager to dispel notions of waning support, emerged from the front entrance of Trump Tower in Manhattan late Saturday afternoon to greet a throng of cheering fans after spending the day huddled with campaign advisers.

For just over a minute, he shook several hands, pumped his fist in the air and waved to the supporters and passersby huddled on a rainy Fifth Avenue before heading back inside.

Asked by reporters if he planned to stay in the race, Trump answered simply: "100%."

But meanwhile, Trump's campaign appeared to be deteriorating around him as the Republican Party descended into a state of chaos.

His own running mate, Indiana Gov. Mike Pence, said he was "offended" by Trump's remarks and canceled plans to represent him at a political event on Saturday.

The third-most powerful Senate Republican, John Thune, a member of Senate Majority Leader Mitch McConnell's inner circle, called on Trump to "withdraw" and let Pence top the Republican ticket. Arizona Sen. John McCain, the party's 2008 nominee, announced he won't vote for Trump.

Vice President Joe Biden condemned Trump on Twitter: "The words are demeaning. Such behavior is an abuse of power. It's not lewd. It's sexual assault."

The words are demeaning. Such behavior is an abuse of power. It's not lewd. It's sexual assault. --Joe — Joe Biden (@JoeBiden) October 8, 2016

Even Trump's wife, Melania Trump, condemned the remarks, saying in a statement released by Trump's campaign, "The words my husband used are unacceptable and offensive to me.

"This does not represent the man that I know," she added. "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."

'They can't make me quit'

Donald Trump, however, was unwavering.

"I'd never withdraw. I've never withdrawn in my life," Trump told The Washington Post Saturday morning. "No, I'm not quitting. I have tremendous support."

He also told The Wall Street Journal there is "zero chance I'll quit."

JUST WATCHED 'Rule 9' outlines rules for replacing GOP nominee Replay More Videos ... MUST WATCH 'Rule 9' outlines rules for replacing GOP nominee 02:18

The comments suggested that Trump was eager to fully turn away from the lewd and sexually aggressive terms he used to describe women in the 2005 conversation and instead pivot toward his political opponents, whoever they may be.

"They're not going to make me quit, and they can't make me quit," Trump told the Post.

He later tweeted, "The media and establishment want me out of the race so badly - I WILL NEVER DROP OUT OF THE RACE, WILL NEVER LET MY SUPPORTERS DOWN! #MAGA"

The media and establishment want me out of the race so badly - I WILL NEVER DROP OUT OF THE RACE, WILL NEVER LET MY SUPPORTERS DOWN! #MAGA — Donald J. Trump (@realDonaldTrump) October 8, 2016

The interviews took place less than 24 hours after previously unaired footage surfaced of Trump bragging about being able to grope women and trying to have sex with a married woman during a 2005 taping for "Access Hollywood."

In that timespan, Republicans have quickly condemned Trump for the comment many of them have deemed "inexcusable" and "indefensible," prompting Trump to issue his first-ever apology of his nearly 16-month campaign.

But Trump also signaled a willingness to fight in the 90-second video statement he posted online just after midnight on Saturday, quickly turning from apologizing to attacking Clinton for her husband's sex scandals.

But the apology did not quell the firestorm.

Several Republicans, including Sens. Mike Crapo of Idaho and Kelly Ayotte of New Hampshire, formally pulled their support of Trump on Saturday morning.

And the Republican National Committee put a temporary pause on its mail operations aiding Trump's campaign in terms of messaging to assess the current situation and decide if they need to change their message on mailings and other get-out-the-vote operations, a top RNC official told CNN.

Republican sources say one option under consideration is ending the joint fundraising agreement between the party and Trump's campaign so the RNC could focus its fundraising exclusively on critical down-ballot races and encourage donors who were opposed to Trump to continue to support the party.

Another Republican source who deals with the joint effort between the campaign and the GOP told CNN it's also under consideration to do a more formal pivot to down-ballot races.

Pence 'cannot defend' remarks

And after a campaign source told CNN that Pence decided not to attend the Wisconsin political event hosted by House Speaker Paul Ryan, Pence released a statement explaining that he was "offended" by Trump's remarks. He also noted he is looking "forward to the opportunity he has to show what is in his heart" at the debate on Sunday.

"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," Pence said. "I do not condone his remarks and cannot defend them. I am grateful that he has expressed remorse and apologized to the American people. We pray for his family and look forward to the opportunity he has to show what is in his heart when he goes before the nation tomorrow night."

JUST WATCHED Trump's lewd 2005 comments stifle campaign Replay More Videos ... MUST WATCH Trump's lewd 2005 comments stifle campaign 05:21

"Wisconsin is off" for Pence, a campaign source told CNN earlier Saturday, adding that the decision was made in the morning.

The source gave no reason for the cancellation, but the Trump campaign had announced Friday evening that Pence would attend the event in Trump's stead after Ryan said he was "sickened" by Trump's comments and announced Trump would no longer attend the event.

Multiple sources told CNN that Trump was asked not to come by Ryan, and one source said the message was delivered via intermediaries.

Ryan declined to discuss the controversy at the event Saturday, though he acknowledged "there is a bit of an elephant in the room."

Trump and Pence spoke Saturday morning, according to a source familiar with the conversation. Another source said Pence told the GOP nominee that the only person who could effectively speak on the matter was Trump himself.

Ready to fight back

Trump indicated he will head into the debate prepared to fight Clinton by raising the marital infidelities of her husband former President Bill Clinton -- a shift since Trump said as recently as Thursday night he would not raise the issue in Sunday's debate.

The real estate magnate spent the day at Trump Tower in New York doing some debate prep, at which former New York City Mayor Rudy Giuliani was present, a source familiar with the situation told CNN.

And Trump argued to the Post that he will be able to weather the comments that have come back to haunt him in part because Clinton, the Democratic nominee, is "so bad" and "so flawed as a candidate."

"Running against her, I can't say it'd be the same if I ran against someone else, but running against her makes it a lot easier, that's for sure," Trump said in the Saturday morning interview.

JUST WATCHED Congressman booed while disavowing Trump Replay More Videos ... MUST WATCH Congressman booed while disavowing Trump 01:08

Trump also told both the Post and the Journal that he is considering delivering remarks Saturday evening to encourage his supporters and address the controversy.

And he also said of the 2005 comments that "people get it. They get life."

Pressed a final time by the Post about the possibility he could quit the race, Trump firmly shut the door on that option.

"Zero chance. I've never quit in my life," Trump said. "I can give you my word that I'm never leaving."