In his floor remarks, McConnell also congratulated Vice-President elect Kamala Harris for the first time.
"I also want to congratulate the vice president-elect, our colleague from California, Senator Harris," he said. "Beyond our differences, all Americans can take pride that our nation has a female vice president-elect for the very first time."
McConnell said while millions wished the election would have yielded a different result, "our system of government has processes to determine who will be sworn in on January the 20th. The Electoral College has spoken."
Trump, even in defeat, holds enormous sway over his party, to the point where many refuse to publicly accept electoral reality or raise any concerns as the President continues to undermine the integrity of US democracy by lying that the election was rigged and stolen from him. House Minority Leader Kevin McCarthy has yet to acknowledge Biden's victory.
Pressed multiple times Monday about whether he'd accept Biden as President-elect, Sen. Steve Daines, a Montana Republican, continued to point to how Trump had won his state's electoral votes.
"Montana cast three electoral votes for President Trump, and the Electoral College voted today," said Daines, who just won reelection in Montana. "And Congress will need to ratify in January."
Still on Monday evening, following the Electoral College process, some GOP members of Congress were signaling they were more willing to accept Biden's victory.
Sen. Roy Blunt of Missouri, a member of Senate GOP leadership who leads the congressional inauguration committee, said they will meet soon to discuss the ceremony to inaugurate Biden on January 20. Just last week, he and other GOP leaders had rejected a Democratic effort to formally note that the committee was preparing for Biden's inauguration.
"We've now gone through the constitutional process and the electors have voted, so there's a President-elect," Blunt said. "With Vice President Biden as the President-elect, the President continues, obviously, to have all the options he has available to him, but the electoral vote today was significant."
Two other members of GOP leadership -- Senate Majority Whip John Thune of South Dakota and Sen. John Cornyn of Texas -- both said that any effort to overturn the election results on January 6, when a joint of session of Congress meets to count the electoral votes, would be fruitless.
"In the end at some point you have to face the music," Thune said Monday on his way to McConnell's office for a leadership meeting. "And I think once the Electoral College settles the issue today, it's time for everybody to move on."
This story has been updated with additional developments Tuesday.
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