Leading Republicans rallied on Monday around President Trump’s refusal to concede the election, declining to challenge the false narrative that it was stolen from him or to recognize President-elect Joseph R. Biden Jr.’s victory even as party divisions burst into public view. Senator Mitch McConnell of Kentucky, the top Republican in Congress, threw his support behind Mr. Trump in a sharply worded speech on the Senate floor. He declared that Mr. Trump was “100 percent within his rights” to turn to the legal system to challenge the outcome and hammered Democrats for expecting the president to concede. In his first public remarks since Mr. Biden was declared the winner, Mr. McConnell celebrated the success of Republicans who won election to the House and the Senate. But in the next breath, he treated the outcome of the presidential election — based on the same ballots that elected those Republicans — as unknown. “President Trump is 100 percent within his rights to look into allegations of irregularities and weigh his legal options,” said Mr. McConnell, the majority leader. “Let’s not have any lectures about how the president should immediately, cheerfully accept preliminary election results from the same characters who just spent four years refusing to accept the validity of the last election.” In Georgia, where the continuing vote count showed Mr. Trump losing the state’s electoral votes, Senators David Perdue and Kelly Loeffler — both Republicans now facing January runoffs to keep their seats — took the extraordinary step of calling on the state’s top election official to resign. Declaring Georgia’s handling of the election an “embarrassment” and citing vague “failures” in an echo of Mr. Trump’s evidence-free charges of stolen votes, they said Brad Raffensperger, the Republican secretary of state, had failed the state. Mr. Raffensperger bluntly rejected their calls, declaring the senators’ claims “laughable” and suggesting that they were merely disgruntled because Mr. Trump might lose and their jobs were on the line. The intraparty feuding underscored how political considerations around Georgia, whose two Senate contests will most likely decide control of the chamber two weeks before Inauguration Day, are driving Republicans’ calculations about how to handle the election results. Republican leaders are reluctant to make any move that might alienate Mr. Trump’s loyal supporters and hurt their candidates’ chances. That includes appearing to bow to the reality that he has lost before the president himself is ready to do so. There was little sign on Monday that would happen in the near term. Mr. Trump’s team rolled out its latest legal moves to challenge the outcome in key states. And in Washington, Emily W. Murphy, a Trump political appointee and administrator of the General Services Administration, refused to formally recognize Mr. Biden as the president-elect with a letter of “ascertainment,” leaving the country’s transition of power in flux. Unperturbed, Mr. Biden plunged ahead with a transition operation that was quickly getting off the ground. In Wilmington, Del., he announced the creation of a Covid-19 advisory board and made an urgent plea to Americans to wear face masks to slow the spread of the coronavirus. He fielded a call from Prime Minister Justin Trudeau of Canada, as a legion of advisers worked quickly to begin lining up candidates to fill top agency posts. “It doesn’t matter who you voted for, where you stood before Election Day,” Mr. Biden said after meeting with the advisory board. “It doesn’t matter your party, your point of view. We can save tens of thousands of lives if everyone would just wear a mask for the next few months.” On Capitol Hill, even as many Republicans privately conceded that the president’s claims were outlandish and mostly avoided repeating them, their public statements suggested that they had no intention of forcing Mr. Trump to accept defeat and begin preparing to hand over the reins of power. They appeared intent on standing by him for a variety of reasons, hoping that the legal process might lend more authority to the final result or that Mr. Trump might simply give in without an intraparty fight. “I think the election is not over until the votes are counted and the legal challenges are decided,” Senator Lindsey Graham, a close ally of the president who was re-elected to his South Carolina seat, told reporters. “That’s why I would encourage the president not to concede.” Only a small group of independent-minded Republicans who have records of breaking with Mr. Trump said they had seen enough. Senator Susan Collins of Maine, a moderate who last week resoundingly won re-election in a state that Mr. Trump lost, congratulated Mr. Biden as the president-elect and stressed the need to begin a transition. Still, even she said that Mr. Trump should be given an opportunity to challenge the results and urged Americans to be patient. “I know that many are eager to have certainty right now,” Ms. Collins said. “While we have a clear direction, we should continue to respect that process.” Mr. Trump and his allies intensified their baseless claims that fraud had wrongly tilted the election in Mr. Biden’s favor, filing a new lawsuit challenging the results in select counties where Mr. Biden won in Pennsylvania. The filing was preceded by a combative press briefing in which the president’s press secretary, Kayleigh McEnany, and the Republican National Committee chairwoman, Ronna McDaniel, alleged that the state election was improperly conducted. Updated Nov. 9, 2020, 7:57 p.m. ET But much of the event included rehashed versions of the arguments Mr. Trump’s team has been making for days, including the false accusation that Democratic-leaning election officials had barred Republican observers from critical counting rooms. At one point, Fox News cut away from the briefing, with the host Neil Cavuto telling his audience, “I can’t in good countenance keep showing you this,” as he noted that Ms. McEnany had not presented any evidence for her charges of Democratic rigging. Earlier, a Michigan court rejected a Trump campaign filing challenging the results in the state, calling the motion “defective” because it lacked several requisite pieces of information, including any evidence. Yet on Capitol Hill, Mr. McConnell and many other Republicans were keeping alive the possibility that Mr. Trump might have legitimate claims. Their approaches were consistent with the way Republicans in Congress have handled Mr. Trump for the last four years, declining to explicitly challenge or contradict the president’s false claims, without necessarily echoing them either. Rather than openly rebuke the false assertion that the election was stolen, Mr. McConnell instead said that “this process will reach its resolution.” “Our system will resolve any recounts or litigation,” he said. But he also took the opportunity to torch Democrats, saying they had no right to expect that Mr. Trump would quickly concede. “At this time last week, small-business owners in cities across America were boarding up their windows in case President Trump appeared to win and far-left mobs decided to reprise their summertime rioting,” Mr. McConnell said. “Suffice to say, a few legal inquiries from the president do not exactly spell the end of the republic.” Democrats were outraged. Following Mr. McConnell on the floor, Senator Chuck Schumer of New York, the minority leader, said flatly that “Joe Biden won this election fair and square.” He called Mr. Trump’s claims “extremely dangerous, extremely poisonous to our democracy” and warned Republican leaders not to give them oxygen. “Republican leaders must unequivocally condemn the president’s rhetoric and work to ensure the peaceful transfer of power,” Mr. Schumer said. A group of 30 former Republican lawmakers, including former Representatives Carlos Curbelo of Florida, Barbara Comstock of Virginia, Tom Coleman of Missouri and Bob Inglis of South Carolina, joined a letter calling on Mr. Trump to concede and accept the results of the election. “We believe the statements by President Trump alleging fraud in the election are efforts to undermine the legitimacy of the election and are unacceptable,” the group wrote. “Every vote should be counted and the final outcome accepted by the participants because public confidence in the outcome of our elections is a bedrock of our democracy.” Few elected Republicans have voiced such views, or even offered the traditional recognition of Mr. Biden’s victory and called for the country to move forward. In her statement Monday, Ms. Collins joined just a handful of House Republicans and just three other Senate Republicans — Mitt Romney of Utah, Lisa Murkowski of Alaska and Ben Sasse of Nebraska — in publicly doing so. “He loves this country, and I wish him every success,” Ms. Collins said in a statement. “Presidential transitions are important, and the president-elect and the vice president-elect should be given every opportunity to ensure that they are ready to govern on Jan. 20.” Other Republicans focused instead on defending what they described as Mr. Trump’s right to pursue legal avenues, although some gently suggested that the time had come for his campaign to substantiate its claims. Pressed on Monday, senators pointed to the 2000 election — whose outcome remained uncertain as a prolonged legal fight reached the Supreme Court — as precedent for withholding a concession as court challenges moved forward. They argued that voters, not the press, decide the election outcome. “There is a process that is available, and I don’t begrudge the president for availing himself of that process,” Senator John Cornyn, Republican of Texas, told reporters on Capitol Hill. “But in the end, they’re going to have to come up with some facts and evidence. But that’s not my job — that’s his campaign’s job.” Jim Rutenberg and Shane Goldmacher contributed reporting.