The measure, authored by Democratic Sen. Joe Manchin of West Virginia, warns, in part, "Whereas any disruption occasioned by the transfer of the executive power could produce results detrimental to the safety and well-being of the United States and its people ..." and ends saying that the Senate "intends that there should be no disruptions by the President or any person in power to overturn the will of the people of the United States."
Trump, meanwhile, stood by his comments first made Wednesday night even as the White House tried to explain what he meant.
As the day began Thursday, the top Republican in the Senate, Majority Leader Mitch McConnell, pushed back on Trump, tweeting, "The winner of the November 3rd election will be inaugurated on January 20th. There will be an orderly transition just as there has been every four years since 1792," but like others did not mention Trump by name.
McConnell made no mention of the explosive subject when he spoke later on the Senate floor.
House Speaker Nancy Pelosi didn't hold back, though, using the president's remarks to urge voter turnout in the 2020 election.
"We don't agonize, we organize, and we want to make sure the American people know how important their vote is," she said. "It's very sad that you even have to ask that question, a real testimony to the need to protect our democracy."
The California Democrat, who has said Trump "admires" leaders like Vladimir Putin of Russia, North Korean dictator Kim Jong Un, and Turkish President Recep Tayyip Erdoğan, called on the president to honor his oath of office.
"He's trying to have the Constitution of the United States swallow Clorox," Pelosi said.
Asked if he will leave the White House peacefully at a White House news conference Wednesday night, Trump had responded, "Well, we're going to have to see what happens. You know that."
When pressed for a second time if he would "commit to making sure that there's a peaceful transferral of power," Trump again turned to his baseless claim of widespread fraud involving mail-in ballots.
"We want to have -- get rid of the ballots and you'll have a very peaceful, there won't be a transfer, frankly," Trump said. "There'll be a continuation."
On Thursday, ABC News Chief White House Correspondent Jonathan Karl asked White House press secretary Kayleigh McNenany, “If the president loses this election, will this White House, will this president assure us that there will be a peaceful transfer of power? It’s a very simple question.”
“The president will accept the results of a free and fair election,” McEnany responded.
When another reporter followed up, asking: “Just to understand this clearly, are the results legitimate only if the president wins?”
"The president will accept the results of a free and fair election,” she repeated. “He will accept the will of the American people.”
Later, speaking to reporters on the White House South Lawn as he left for North Carolina, Trump would not say whether he would consider the election legitimate only if he wins.
“We want to make sure the election is honest. I’m not sure that it can be,” Trump said.
The Republican who heads the Senate Judiciary Committee, Lindsey Graham, told "Fox and Friends" Thursday morning there was nothing to worry about.
"People wonder about the peaceful transfer of power. I can assure you it will be peaceful. Now, we may have litigation about who won the election ... and if Republicans lose we'll accept the result," Graham said, noting a court challenge was a key reason why Trump's Supreme Court nominee needed to be confirmed before the election.
GOP Sen. Mitt Romney of Utah was more pointed in his reaction in a tweet overnight, saying a peaceful transition of power was "fundamental to democracy," but he also did not call out Trump by name.
On Thursday, Romney said he was "absolutely confident there will be a peaceful transition if there’s a new president," when asked by reporters.
"No question that all the people sworn to support the Constitution would assure that there would be a peaceful transition of power, including the president," Romney said.
Rep. Liz Cheney, the third-ranking House Republican who has pushed back at Trump previously, tweeted a peaceful transfer of power is "fundamental to the survival of our Republic."
Several other Republicans pointed to the vital role a peaceful transfer of power plays in a democratic system, including Sen. Susan Collins, R-Maine, who said she was concerned by Trump's comments.
"The peaceful transfer of power is a fundamental tenet of our democracy. And I am confident that we will see it occur once again," Collins said Thursday. "I don't know what his thinking was, but we have always had a peaceful transition between administrations."
In an effort to deflect criticism of Trump's remarks, several Republicans pointed to comments Hillary Clinton made in August to Showtime's "The Circus." Clinton, after outlining that she believes Trump may target absentee ballots to discredit the election outcome, advised that Biden "not concede."
"Joe Biden should not concede under any circumstances because I think this is going to drag out, and eventually I do believe he will win if we don't give an inch and if we are as focused and relentless as the other side is," Clinton said in the interview.
"What he says doesn't matter any more than what Hillary Clinton advised Biden to do: Don't concede the election," Senate Finance Committee Chairman Chuck Grassley told reporters Thursday.
Sen. Roy Blunt, R-Mo., also pointed to the Clinton comment.
"Whether it’s Mrs. Clinton telling Joe Biden to never concede or any other suggestion that somehow we won’t finish this election year appropriately of course I’m concerned about that but both sides are unhelpful on this topic," Blunt said. "But there will be a - if the president is reelected there is no transition and if he’s not there will be a peaceful transition on January 20."
Sen. Tim Kaine, D-Va., said Thursday he is not at all concerned that Biden would not accept the outcome of the 2020 election.
"If it's a clear outcome, there is no question," Kaine said. "We could have a murky outcome and then we have to wait and get it to be clear but there is no question that he will accept a clear outcome."
The second-ranking Democrat in the Senate, Dick Durbin of Illinois, called Trump "a real danger to the Constitution."
"This is frightening rhetoric from the President of the United States. What he said to the American people was, go ahead and vote if you wish. But I'm not sure I’ll accept your vote as the real sentiment of America. That questions the very foundation of our democracy, if a president does not accept the results of an election. It's not the first time he’s said it, so we know that he’s convinced of how important that statement is. But the American people should be convinced that this president is a real danger to our Constitution," he told reporters Thursday morning.
His comments were echoed by Sen. Elizabeth Warren, D-Mass., who likened the president to a dictator.
"Donald Trump’s statement is what a dictator says although usually dictators don’t announce in advance what their plans are," Warren said. "He wants to be named president for life, king of the country. That’s not how Democracy works."
GOP Sen. Kevin Cramer of North Dakota wasn't concerned.
"The president speaks in overly extreme manners on occasion, I didn't find what he said last night to be overly extreme quite honestly. I just thought that what he's making the point that we'll see what happens after the election," Cramer said.
ABC News' Ben Gittleson, Elizabeth Thomas, Ben Siegel and Trish Turner contributed to this report
This report was featured in the Friday, Sept. 25, 2020, episode of “Start Here,” ABC News’ daily news podcast.
"Start Here" offers a straightforward look at the day's top stories in 20 minutes. Listen for free every weekday on Apple Podcasts, Google Podcasts, Spotify, the ABC News app or wherever you get your podcasts.