President Trump’s bellicose pledge to fight the outcome of the election in the courts crashed on Friday into skeptical judges, daunting Electoral College math and a lack of evidence for his claims of fraud.
On a day that began with vote tallies in Georgia and Pennsylvania tipping in Joseph R. Biden Jr.’s favor, Mr. Trump’s campaign declared, “This election is not over,” as the Republican National Committee announced it had activated “legal challenge teams” in Arizona, Georgia, Michigan and Pennsylvania. And the Trump forces named a new general to lead the effort, the hardened conservative political combatant David Bossie.
But none of the dozen or so lawsuits they had brought in battleground states appeared to be gaining any traction in the courts. And in any case, none seemed likely to give Mr. Trump the edge he would need in vote counts in the states that will determine the outcome.
In seeking to foment widespread doubt about the legitimacy of the election, Mr. Trump and his surrogates seemed less focused on substantive legal arguments that could hold up in court than on bolstering the president’s political narrative, unsupported by the facts, that he was somehow being robbed of a second term.
The most high-profile step of the day came when Pennsylvania Republicans asked the U.S. Supreme Court to step in and require election officials in the state to segregate ballots that arrived after Election Day and not to include them for now in the vote totals in the largest and most critical of the swing states.
On Friday evening, Justice Samuel A. Alito Jr. agreed to the request.
But the move was almost entirely for show: Pennsylvania is already segregating those ballots, counting them separately and not including them in the announced vote totals. The secretary of state, over the objections of Republicans and Mr. Trump, has said they can be counted if they arrived by 5 p.m. on Friday, in line with a state court ruling that the Supreme Court has left open the possibility of reviewing again.
A state official said the ballots in question number in the thousands but not tens of thousands.
Their lack of progress in stopping the count or making a persuasive case for large-scale ballot fraud left Mr. Trump and his team increasingly reliant for political salvation on recounts — which appeared likely to take place in Georgia, Nevada, Arizona and Wisconsin but which rarely result in big swings in vote counts.
The Trump effort may be getting a boost from state legislatures in Pennsylvania and Wisconsin, which are both controlled by Republicans. In Wisconsin, Robin Vos, the speaker of the State Assembly, directed a legislative committee to “use its investigatory powers” to conduct a review of the election, again raising the specter of voter fraud without offering specific evidence.
In Pennsylvania, the two top Republicans in the legislature called on Gov. Tom Wolf, a Democrat, to conduct “an immediate audit” of the election.
Officials with the Biden campaign said they would meet every legal challenge Mr. Trump brought but said they were confident that none of the cases they had seen so far seemed likely to loosen Mr. Biden’s tightening grasp on the presidency.
“The Republican legal claims are utterly baseless and have failed and will continue to fail in the courts,’’ said Bob Bauer, a senior adviser to the Biden campaign. “They serve no purpose other than to echo Donald Trump’s discredited and shameful attack on the democratic process.”
In a statement, the Republican Party’s chief counsel, Justin Riemer, said, “We are focused on protecting the integrity of the vote and ensuring all legally cast ballots are counted,” but he was not made available for further comment on the particulars of the party’s legal strategy.
The intensive Trump effort was the endgame of a long-planned contingency in which the president planned to challenge any possible loss through claims that the voting system was “rigged” against him. The campaign has been abetted by a robust pro-Trump media ecosystem, with Mr. Trump’s own social media accounts at its center, amplifying his false claims.
As the legal push has failed to make substantial headway this week it has taken on an urgent, at times desperate, tone.
On Thursday, Eric Trump, one of the president’s sons, said on Twitter, “I truly hope the @FBI/@DOJ engages immediately.”
The former Republican Speaker of the House, Newt Gingrich, echoed the president’s son on Fox News, calling for the jailing of poll workers and more involvement from Attorney General William P. Barr.
But in the days since the election, Mr. Barr and the Justice Department have been largely silent. After initially echoing Mr. Trump’s warnings about electoral fraud, Mr. Barr muted his statements, and Mr. Trump complained to aides about the department’s lack of action.
But a supportive outside group, True the Vote — one of the most prominent promoters of the false narrative that “voter fraud” is rampant in the United States — sought to help Mr. Trump build his cases. On Friday, it announced it had formed a $1 million “Whistleblower Defense Fund” to “incentivize” witnesses to step forward with charges of malfeasance.
No state was seeing more legal activity than Pennsylvania, where the Trump campaign and local Republicans had brought at least a half-dozen lawsuits immediately before and after Election Day.
The case that had the potential to affect the most votes was the one filed at the Supreme Court on Friday seeking an order that would force election officials in Pennsylvania to separate mailed ballots that arrived after Election Day from other mailed ballots. In its emergency application, the party acknowledged that the secretary of state’s office had already ordered election officials to do so but said it had no way to know if they had complied.
Republicans have been unsuccessfully fighting the secretary of state’s decision to allow election officials in Pennsylvania to count mail-in ballots that arrived at their offices by Friday as long as they had postmarks from Election Day or before. The Supreme Court had twice passed up opportunities to rule on the dispute, though the case is still technically pending, giving the justices the opportunity to weigh in if they saw reason to do so.
But even if the court were to take the case and rule in favor of the Republicans to wipe out all of the ballots in question — votes from mail ballots have overwhelmingly gone to Mr. Biden — it would not affect the current vote totals, which do not include the ballots that came in after Election Day. By early Friday evening, Mr. Biden had a lead of about 17,000 votes in Pennsylvania.
Other suits in Pennsylvania sought to knock out votes that were the result of a decision by Secretary of State Kathy Boockvar to allow county officials to give voters a chance to fix mistakes in their rejected mail-in ballots or to cast provisional votes instead.
A federal judge dismissed one of those cases on Friday, in Montgomery County outside Philadelphia, two days after implying during a hearing that he viewed the effort as a move to disenfranchise voters who were following state instructions.
But even if that case had succeeded, it would have affected only 93 votes. Similarly, in Michigan, a judge dismissed a Republican suit challenging the vote count in the state, noting the counting was already effectively over and dismissing some of the evidence as based on hearsay.
Frustrated supporters of the president like the talk radio host Mark Levin called on Republican legislatures in states including Pennsylvania to use their constitutional authority to send a pro-Trump delegation of electors to the Electoral College regardless of the popular vote.
Asked during a news briefing on Friday if state Republicans would do so, the majority leader of the Pennsylvania senate, Jake Corman, replied, “We want to stay in the tradition that the popular vote winner wins the election.”
Even if the state Republicans could make such a move legally — the legislature “cannot simply ignore the popular vote,” said Attorney General Josh Shapiro of Pennsylvania — it was an extreme example of the lengths to which some Trump supporters seem willing to go.
In several states Mr. Trump was challenging state voting rules, effectively seeking to nullify votes cast in accordance with official guidelines.
Republicans in Nevada asked the Justice Department late Thursday to open an investigation into voter fraud involving 3,000 ballots, a move Democrats said was based on “entirely fabricated” claims.
The Justice Department declined to address the matter publicly, but its guidelines generally preclude opening criminal investigations into election-related matters until results are completed.
Local Republicans included the allegation in a suit they filed in federal court seeking a change in the way Clark County, Nev., is conducting its count. A judge dismissed the case on Friday evening, when the vote count showed a 22,000-ballot lead for Mr. Biden.
Michael S. Schmidt, Katie Benner and Adam Liptak contributed reporting.