Lee called on Trump to abandon his campaign, saying it was time for the Republican Party to "expect more".
Trump apologized in a videostatement released by his campaign early Saturday morning.
Alaska Sen. Lisa Murkowski said Trump has "forfeited the right to be our party's nominee".
But Saturday, he said, "I and other Republican colleagues are starting to wonder if this is really the standard-bearer we want".
He also bragged about his ability to "do anything" with women, even groping them.
West Virginia Sen. Shelly Moore Capito but notes that the "appropriate next step may be for him to reexamine his candidacy".
And some voters prefer neither candidate, choosing to support Libertarian nominee Gary Johnson, Green Party nominee Jill Stein, or independent Evan McMullin.
But these new revelations come at a time when the White House race seems to be slipping away from Trump. "Ever", said Reince Priebus, who had stood by Trump through his past provocative comments. "And I think that's a good thing".
The Republican nominee and Democratic rival Hillary Clinton are scheduled to meet for a second debate on Sunday in St. Louis. More than 60 prominent Republican current and former officeholders issued statements condemning Trump's remarks about women, including more than 20 who called for him to end his presidential bid. "We can not allow this man to become president".
In the video from 2005, obtained and released by The Washington Post, Trump uses extraordinarily vulgar and predatory language as he describes hitting on a married woman and grabbing women's crotches.
The top Republican woman in Congress, Rep. Cathy McMorris Rodgers, R-Washington, rebuked Trump's comments and the advances he described on video. "It's like a magnet".
"I just start kissing them", he said. "You know I'm automatically attracted to handsome - I just start kissing them". You can do anything. "Grab them by the pussy".
Trump has vehemently said he won't drop out.
He continued, "And I moved on her very heavily".
House Speaker Ryan said Friday was "sickened" by Trump's remarks. I'll admit it. I did try and f- her. In the first debate, Clinton chided Trump for not releasing his tax returns and speculated that he might not be paying any federal taxes. "I was younger, less mature and acted foolishly in playing along".
"Access Hollywood" said a recent Associated Press story about Trump's lewd behind-the-scenes comments as star of "The Apprentice" led it to dig through its archives and turn up the previously unaired tape.
Watch the interplay between the moderators and Trump, who has complained about unfair treatment of the GOP ticket by the moderators of both the first presidential debate and the vice presidential faceoff.
"Nothing indefensible that Donald Trump said 20 years ago is going to change that", he said. He also called the footage of his comments a "distraction" from bigger issues at stake in the November 8 election. It's not just that we have further evidence of what sort of person Trump is. Sullivan has been joined by Ron Klain, who advised President Barack Obama; Karen Dunn, a longtime lawyer and aide; John Podesta, Clinton's campaign chairman; and Jennifer Palmieri, Clinton's communications director. The Republican presidential candidate demonstrated extreme chauvinism in the 2005 interview where Trump also talked about trying to sleep with married women, and blatantly stated he can sexually assault women because he is a "celebrity".
"The comments made by Mr. Trump were disgusting and totally unacceptable under any circumstance".
Senator Mark Kirk, who is up for re-election in a tight race in the state of IL, also tweeted that Trump should drop out of the race and the Republican Party "should engage rules for emergency replacement". By NBC New's count, 38 Republicans have condemned their party's presidential nominee and almost a dozen have called for him to drop out.
Republican Darryl Glenn, who's running against incumbent Democratic Sen.
But, Hillary Clinton has said that Trump should not be President after what he has said.
But major damage was already done, with several fellow Republicans withdrawing their endorsements of Trump. "I don't even wait", Trump, said, as Bush eggs him on. "We don't know where his floor is now". Trump and Bush speak candidly, as if they didn't know their microphones were recording, but the content of the discussion is what's most shocking of all.