(CNN) — Senate Majority Leader Mitch McConnell for the first time acknowledged Joe Biden’s victory and referred to him as President-elect, six weeks after Election Day and amid President Donald Trump’s continued refusal to accept defeat.

“The electoral college has spoken,” McConnell said in remarks from the Senate floor in the US Capitol, adding, “Today I want to congratulate President-elect Joe Biden.”

McConnell’s comments are significant given that many Senate Republicans still wouldn’t recognize Biden’s victory on Monday evening even after the Electoral College made the win official.

Behind the scenes, McConnell also urged Republican senators on a private conference call not to join House members on January 6 to object to state electoral results, a source on the call tells CNN, when Congress formally counts the Electoral College votes and members of Congress can object. Other top Republicans — including Senate Majority Whip John Thune and Senate Rules Chairman Roy Blunt — echoed that sentiment. Doing so, they said, would be fruitless and force them to cast a politically challenging vote against the President that day.

Democrats have criticized Republicans’ refusal to accept the outcome of the Election, and highlighted how it took more than 40 days since Americans went to the polls for many GOP members of Congress to speak up.

“The fact that it took six weeks for my colleagues to recognize reality and stop undermining our Democratic process is sad and disappointing,” said Senate Democratic Whip Dick Durbin in a floor speech Tuesday following McConnell’s remarks.

LIVE UPDATES: The latest on Biden’s transition

Senate Minority Leader Chuck Schumer said Tuesday that Trump needs to “take his cue from Leader McConnell, that it’s time to end his term,” in a floor speech and also called on his GOP colleagues to follow suit.

“Our Republican colleagues, for the sake of our democracy, for the sake of the peaceful transition of power, should stop the shenanigans,” Schumer said “Stop the misrepresentations and acknowledge that Joe Biden will be our next president.”

McConnell, a Kentucky Republican, made his comments after praising the Trump presidency in the past tense, touting his administration’s accomplishments, including the country’s “economic prosperity,” “foreign policy,” judicial appointments, and “bold regulatory changes” in a floor speech.

“It would take far more than one speech to catalog all the major wins the Trump administration has helped deliver for the American people,” he said. “The outsider who swore he would shake up Washington and lead our country to new accomplishments, both at home and abroad proceeded to do exactly that. President Trump and Vice President Mike Pence deserve our thanks and our gratitude for their tireless work and their essential roles in all these victories and in many more.”

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.”

Following McConnell’s remarks, Republican Sen. Rick Scott of Florida, who is in GOP leadership, would not say if Biden is President-elect, when asked by CNN. The Florida Republican did say, however, that the Electoral College has voted and “there will be a smooth transition of power on the 20th of 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.

Blunt 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 — Thune and Sen. John Cornyn of Texas — both said Monday that any effort to overturn the election results on January 6 would not be successful.

“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.”

On Tuesday, Cornyn called the Electoral College vote “a pivotal moment.”

“I think everybody realized it yesterday that counting the voting of the electors was a pivotal moment, and I agree,” Cornyn told reporters Tuesday, when asked if he recognizes Biden as President-elect.

Asked whether Trump should concede, or congratulate Biden, Cornyn said he isn’t going to give any advice to the President, but “I think he will do the right thing.”

This story has been updated with additional developments Tuesday.

The-CNN-Wire™ & © 2020 Cable News Network, Inc., a WarnerMedia Company. All rights reserved.