Democrats and reproductive rights activists pressed the Biden administration on Tuesday to take more aggressive action to stop a Texas law that prohibits abortions after six weeks of pregnancy, even as administration officials and legal experts acknowledged it would be difficult to curtail the law in the coming months.

House Democrats, following similar calls over the weekend from a leading liberal legal scholar, pushed Attorney General Merrick B. Garland to use the Justice Department’s powers to prosecute Texas residents now empowered under the law to sue women seeking abortions.

“We urge you to take legal action up to and including the criminal prosecution of would-be vigilantes attempting to use the private right of action established by that blatantly unconstitutional law,” the House Judiciary Committee chairman, Representative Jerrold Nadler of New York, and 22 other House Democrats wrote in a letter to Mr. Garland.

The Justice Department referred reporters seeking a response to the letter back to a statement by Mr. Garland a day earlier, in which he said that law enforcement officials were “urgently” exploring options to challenge the Texas law “in order to protect the constitutional rights of women and other persons, including access to an abortion.”

The demands by House Democrats were the latest push from liberals after the Supreme Court decided last week to allow the Texas law to take effect. Instead of using the law enforcement powers of the state, the law gives private citizens the right to sue anyone who helps a woman obtain an abortion. Under the law, those plaintiffs can win $10,000 and recover their legal fees if they win.

The law has emerged as the starkest example of how former President Donald J. Trump tipped the balance of the Supreme Court to the right by appointing three conservative justices.

Now, President Biden’s base is pushing for him to do more. But because of the novel way the Texas law was written, the conservative majority on the Supreme Court and the slow pace of the judicial system, Biden administration officials have few options to protect abortion rights in Texas in the short-term.

“The Department of Justice has few legal avenues likely to succeed, and the federal courts are not likely to be receptive to their challenges,” said Elizabeth W. Sepper, a law professor at the University of Texas at Austin.

Mr. Biden signaled his outrage last week by calling the law “almost un-American” and ordered the Departments of Justice and Health and Human Services to find ways to ensure women can safely seek abortions in Texas, a task that administration officials say will take some time and creativity.

Mr. Garland said in his statement on Monday that the federal government would beef up its enforcement of a 1994 law designed to protect women from harassment and intimidation as they sought abortions.

“The department will provide support from federal law enforcement when an abortion clinic or reproductive health center is under attack,” Mr. Garland said. “We have reached out to U.S. Attorneys’ Offices and F.B.I. field offices in Texas and across the country to discuss our enforcement authorities.”

In the face of the calls by Democrats for the administration to do more, the White House and the Justice Department declined to say on Tuesday what else they might have in store.

“The White House Counsel’s Office, the Justice Department, the Department of Health and Human Services are continuing to look for ways to expand women’s access to health care,” the White House press secretary, Jen Psaki, told reporters.

Senator Sheldon Whitehouse, Democrat of Rhode Island, also called on Democrats to investigate whether the Texas law was part of a national campaign being waged by conservative groups and funded by unnamed donors that was intended to push certain legislation, like voter suppression laws.

“We have done a rotten job at exposing that,” Mr. Whitehouse said. “We have been negligent, not just weak, in letting this transpire and not doing the work to tell the American public about it.”

The idea of using the prosecutorial powers of the Justice Department to take on the Texas law gained traction this weekend through an opinion essay in The Washington Post by the constitutional scholar Laurence H. Tribe. The best way for Democrats to protect abortion rights is for Congress to pass a law, Mr. Tribe argued. But he said that Democrats likely do not have enough votes in Congress and warned that the Supreme Court could overturn such a law anyway.

Instead, Mr. Tribe said, Mr. Garland has the power to take legal action against those who seek to deprive someone of their constitutional rights.

Mr. Tribe said that the law Mr. Garland needed to use had been passed in the years after the Civil War to stop members of the Ku Klux Klan from lynching Black people and trying to stop them from voting.

“The attorney general should announce, as swiftly as possible, that he will use federal law to the extent possible to deter and prevent bounty hunters from employing the Texas law,” Mr. Tribe wrote. “If Texas wants to empower private vigilantes to intimidate abortion providers from serving women, why not make bounty hunters think twice before engaging in that intimidation?”

But Brian Fallon, the executive director of the progressive group Demand Justice, which advocates expanding the number of seats on the Supreme Court, called on Democrats to focus on that larger target, saying it could affect myriad policy issues.

“That’s a problem that is much bigger and harder to solve, and a lot of people continue to avoid it all together,” Mr. Fallon said. “The current reality is there will be further innovations beyond the Texas statute that we can expect in the months and years to come.”