The Supreme Court on Friday rebuffed a last-ditch attempt by Texas to block electors from four battleground states — all of which backed President-elect Joe Biden — from voting in the Electoral College, delivering a fatal blow to President Trump and his allies in their quest to overturn the results of the presidential election.

The high court refused to take up the lawsuit filed Monday by Texas Attorney General Ken Paxton that took aim at the election results in Pennsylvania, Michigan, Georgia and Wisconsin.

“Texas has not demonstrated a judicially cognizable interest in the manner in which another State conducts its elections. All other pending motions are dismissed as moot,” the court said in an unsigned order.

But Justice Samuel Alito, joined by Justice Clarence Thomas, said the court does “not have discretion to deny the filing of a bill of complaint in a case that falls within our original jurisdiction. I would therefore grant the motion to file the bill of complaint but would not grant other relief, and I express no view on any other issue.”

The decision by the Supreme Court not to wade into the dispute marks the second loss for Mr. Trump and his GOP backers this week. On Tuesday, the justices rejected a bid from Pennsylvania Republicans to overturn the results of the election there.

The long-shot lawsuit from Paxton, which was supported by 17 state attorneys general and more than 100 House Republicans, joined a flurry of legal challenges brought by Mr. Trump and his attorneys that have unsuccessfully sought to reverse the outcome of the presidential election in battleground states won by Mr. Biden. Federal and state judges have largely tossed out their lawsuits because they lacked evidence to support claims the election was rife with fraud.

Still, Mr. Trump has asserted the Supreme Court, with a 6-3 conservative majority, would come to his rescue and help deliver him a second term in the White House.

“Now that the Biden Administration will be a scandal plagued mess for years to come, it is much easier for the Supreme Court of the United States to follow the Constitution and do what everybody knows has to be done. They must show great Courage & Wisdom. Save the USA!!!” he tweeted Friday.

The high court, however, has declined to intervene in cases challenging the results of the election.

In his lawsuit filed directly with the Supreme Court against the states, Paxton claimed government officials used the coronavirus pandemic as justification to change their election rules, which led to a “weakening of ballot security.” The officials, he claimed, “flooded” their states with mail-in ballots and ballot applications, making “the 2020 election less secure.”

Officials in Michigan, Pennsylvania, Wisconsin and Georgia “usurped their legislatures’ authority and unconstitutionally revised their state’s election statutes,” Paxton alleged. “They accomplished these statutory revisions through executive fiat or friendly lawsuits.”

In his complaint, Paxton asked the Supreme Court to extend the December 14 deadline for the Electoral College to meet and block the states from certifying their presidential electors or having them cast their votes for president in the Electoral College. If the states have already appointed their presidential electors using the 2020 election results, he called on the high court to direct state legislatures to appoint a new set of electors.

But the top law enforcement officials from each of the four states urged the Supreme Court not to take up the dispute, separately noting the claims raised in Texas’s complaint have already been rejected by federal and state courts.

Pennsylvania Attorney General Josh Shapiro called the efforts to overturn the election results “legally indefensible” and “an afront to principles of constitutional democracy.”

“Texas’s effort to get this court to pick the next President has no basis in law or fact,” he wrote in a filing with the Supreme Court. “The court should not abide this seditious abuse of the judicial process, and should send a clear and unmistakable signal that such abuse must never be replicated.”

Shapiro told the court Texas waited to request an injunction to invalidate Pennsylvania’s election results “because all of the other political and litigation machinations of petitioner’s preferred presidential candidate have failed.”

“The Trump campaign began with a series of meritless litigations. When that failed, it turned to state legislatures to overturn the clear election results. Upon that failure, Texas now turns to this Court to overturn the election results of more than 10% of the country,” he continued. “Texas literally seeks to decimate the electorate of the United States.”

Mr. Trump filed a motion with the Supreme Court on Wednesday asking to join Texas’ case, echoing its claim that elections officials “made a systematic effort to weaken measures to ensure fair and impartial elections by creating new rules for the conduct of the elections” under the guise of the pandemic.

“These new rules were aimed at weakening, ignoring, or overriding provisions of state law that are aimed at ensuring the integrity of the voting process,” Chapman University law professor John Eastman, the president’s lawyer, wrote in a filing with the court.

Mr. Trump alleged it “should come as no surprise” that many Americans believe the election was stolen — as he has been claiming near-daily — because no presidential candidate has ever lost the election after winning both Florida and Ohio, as he did. The president, however, is incorrect: Richard Nixon won both states in 1960 but lost the presidential election.

Republican attorneys general from 17 states filed a friend-of-the-court brief with the Supreme Court in support of Texas’s lawsuit, and GOP Congressman Mike Johnson of Louisiana, joined by 105 House Republicans, are also backing Texas’s efforts before the high court.

A slew of other states have also jumped into the fray — 19 states and the District of Columbia joined the four battleground states in urging the Supreme Court to reject Texas’s attempt to toss out the election results, while Ohio said it does not support Paxton’s proposed relief.

Ordering state legislatures to appoint its own electors would violate the Constitution, Ohio Attorney General Dave Yost, a Republican, and Solicitor General Benjamin Flowers told the court.

“Federal courts, just like state courts, lack authority to change the legislatively chosen method for appointing presidential electors. And so federal courts, just like state courts, lack authority to order legislatures to appoint electors without regard to the results of an already-completed election,” they argued. “What is more, the relief that Texas seeks would undermine a foundational premise of our federalist system: the idea that the states are sovereigns, free to govern themselves.”

Paxton’s effort to overturn the election results was derided by top officials from the states named in his suit, who called it a waste of taxpayer dollars. Several Republican lawmakers were also skeptical Paxton would succeed.

Republican Senator John Cornyn of Texas told reporters Wednesday the lawsuit was “unprecedented” and noted states oversee their own election laws and voting, while GOP Senator Mitt Romney of Utah called the effort “simply madness.”

“The idea of supplanting the vote of the people with partisan legislators is so completely out of our national character that it’s simply mad,” he told reporters. “Of course the president has the right to challenge results in court, to have recounts. But this effort to subvert the vote of the people is dangerous and destructive of the cause of democracy.”

The four states named in Paxton’s legal challenge have already certified their election results, formalizing Mr. Biden’s victory over Mr. Trump, and the clock for any relief for the president is quickly running out.

December 8 was the “safe harbor” deadline for states to resolve election disputes and certify its election results before the Electoral College meets Monday.