WASHINGTON — The Supreme Court on Tuesday refused to block a ruling from a federal judge in Texas requiring the Biden administration to reinstate a Trump-era immigration program that forces asylum seekers arriving at the southwestern border to await approval in Mexico.

The court’s brief unsigned order said that the administration had appeared to act arbitrarily and capriciously in rescinding the program, citing a decision last year refusing to let the Trump administration rescind the Obama-era program protecting the young immigrants known as dreamers.

The court’s three more liberal members — Justices Stephen G. Breyer, Sonia Sotomayor and Elena Kagan — said they would have granted a stay of the trial judge’s ruling. They did not give reasons. The case will now be heard by an appeals court and may return to the Supreme Court.

The challenged program, known commonly as Remain in Mexico and formally as the Migrant Protection Protocols, applies to people who left a third country and traveled through Mexico to reach the U.S. border. After the policy was put in place at the beginning of 2019, tens of thousands of people waited for immigration hearings in unsanitary tent encampments exposed to the elements. There have been widespread reports of sexual assault, kidnapping and torture.

President Biden suspended and then ended the program. Texas and Missouri sued, saying they had been injured by the termination by having to provide government services like drivers’ licenses to immigrants allowed into the United States under the program.

On Aug. 13, Judge Matthew J. Kacsmaryk of the U.S. District Court for the Northern District of Texas, in Amarillo, ruled that a federal law required returning noncitizens seeking asylum to Mexico whenever the government lacked the resources to detain them.

That was a novel reading of the law, the acting solicitor general, Brian H. Fletcher, told the justices. That view had “never been accepted by any presidential administration since the statute’s enactment in 1996,” including the Trump administration, he said.

Judge Kacsmaryk suspended his ruling for a week, and the U.S. Court of Appeals for the Fifth Circuit, in New Orleans, refused to give the administration a further stay while it pursued an appeal, prompting an emergency application for a stay in the Supreme Court. On Friday, shortly before the ruling was to go into effect, Justice Samuel A. Alito Jr. issued a short stay to allow the full Supreme Court to consider the matter.

The Supreme Court has had previous encounters with the program. In response to an emergency application from the Trump administration, the court revived the program last year after a federal appeals court blocked it.

The justices later agreed to hear the Trump administration’s appeal from a decision of the U.S. Court of Appeals for the Ninth Circuit, which had blocked the law, saying it was at odds with federal law and international treaties and was causing “extreme and irreversible harm.” But the Supreme Court dismissed the case in June in response to a request from the Biden administration.

Mr. Fletcher urged the justices to give the Biden administration the same deference it had afforded the Trump administration.

“In recent years, this court has repeatedly stayed broad lower-court injunctions against executive branch policies addressing matters of immigration, foreign policy and migration management,” he wrote. “It should do the same here.”

He added that the executive branch had broad authority over immigration. “The district court’s injunction,” Mr. Fletcher wrote, “effectively dictates the United States’ foreign policy.”