LAREDO, Texas — When Representative Henry Cuellar stepped onstage at a campaign rally in San Antonio this week, he spoke of education, health care and his experience in Congress. But as Mr. Cuellar, a nine-term Democratic congressman, faces his toughest re-election challenge yet, one word did not escape his lips: abortion.

Mr. Cuellar, the most staunchly anti-abortion Democrat in the House, will face a primary runoff later this month against Jessica Cisneros, a 28-year-old immigration lawyer and a progressive supporter of abortion rights.

Democrats across the country hope the leaked draft Supreme Court opinion overturning Roe v. Wade will help galvanize their voters in the midterm elections, potentially rallying support for the party in a year that is widely expected to benefit Republicans.

But one of the first tests of just how much the issue will motivate Democratic voters will come in a primary runoff at the end of May, in the South Texas district held by Mr. Cuellar where conservative Democrats have regularly succeeded. It is a heavily Latino district that includes the border city of Laredo and parts of San Antonio, where Catholicism dominates, and where, as Mr. Cuellar showed from the stage, the subject of abortion often goes undiscussed.

Dr. Enrique Benavides III, a 51-year-old obstetrician-gynecologist who manages a women’s health clinic with his father in Laredo, said that typically when patients request abortions, he gives them information about a clinic in San Antonio, more than a two-hour drive away.

Dr. Benavides described himself as a pro-choice Catholic Democrat who will vote for Mr. Cuellar. “Democrats here are very different than those on the coasts, very Catholic and conservative,” he said.

Abortion rights advocates believe the Supreme Court’s draft ruling will shake up the race, providing a helpful jolt for Ms. Cisneros. But supporters of Mr. Cuellar and some local Democratic officials say the district’s voters, who lean socially conservative, are unlikely to be moved by the issue. And several national Democratic leaders who have publicly made fiery vows to maintain abortion rights are nonetheless standing behind Mr. Cuellar, with some warning that a win for Ms. Cisneros later this month could give Republicans an edge in November.

In interviews with nearly a dozen men and women in Laredo, most said they would support Mr. Cuellar in spite of — or even because of — his stance on abortion.

These voters, several of whom expressed support for abortion rights but said they would still vote for an anti-abortion congressman, revealed the uphill battle Ms. Cisneros faces as she tries to convince voters to oust a familiar political figure whose family has long been a fixture in the community. Even among Democrats, support for abortion rights may not uniformly motivate voters, particularly working-class Latinos — a demographic that has shown signs of drifting away from the party.

Hector Gomez, 67, has known Mr. Cuellar since they were classmates at J.W. Nixon High School in the 1970s, and has voted for the congressman in every election since he first ran in 2006.

“He’s doing his job,” said Mr. Gomez, an antique store owner, adding that although he is Catholic and opposed to abortion, the issue does not determine his vote. “Mr. Cuellar is the best choice because he’s not someone you can just brush off.”

Texas’ 28th Congressional District stretches from the Mexican border to San Antonio, and Laredo is its political center. A working-class city, it has been a Democratic stronghold for decades but remains culturally conservative, with residents who fill Catholic church pews on Sundays. Many describe themselves as apolitical, and said they are more focused on making ends meet than staking out positions on partisan political issues.

Before this week’s leaked Supreme Court opinion, abortion had not been the central issue in the primary campaign, though several national abortion rights groups had invested heavily in the district, focusing on the new state abortion restrictions in Texas. Ms. Cisneros did not run a television advertisement on the issue until late last month, according to AdImpact, an ad tracking firm. Until the focus on abortion was renewed this week, the runoff had been a mostly sleepy affair, with observers predicting an extremely low turnout.

Now, Ms. Cisneros and her supporters have moved to use the threat to abortion rights as a primary motivator for both voters and donors.

“We’re really at a moment where people are fired up and they know how much they are at risk of losing,” said Kristin Ford, the vice president of communications and research at the abortion rights group NARAL Pro-Choice America, which has sent organizers to Laredo to campaign for Ms. Cisneros.

Ms. Cisneros argues that the district is not nearly as conservative as Mr. Cuellar and his backers suggest, and that attitudes are changing.

“This ignited the urgency,” she said in an interview on Friday. “When we defeat the anti-choice Democrat, that’s going to set the tone for the rest of the midterms that we want a pro-choice Democratic majority in power.”

Several prominent left-wing lawmakers, including Representative Alexandria Ocasio-Cortez of New York, have made fund-raising appeals for Ms. Cisneros. Earlier this week, Senator Elizabeth Warren of Massachusetts, one of Ms. Cisneros’s most prominent backers, used an appearance on MSNBC to make a full-throated fund-raising pitch.

“If you’re mad when you listen to this, send Jessica Cisneros ten bucks,” she said.

And during a recent virtual fund-raiser for Ms. Cisneros, Ms. Warren, speaking of Mr. Cuellar, said, “When it comes to reproductive rights, Henry’s got a record that makes my blood boil,” according to prepared remarks obtained by The New York Times.

Ms. Cisneros first challenged Mr. Cuellar in 2020, when she lost the Democratic primary by 4 percentage points, and quickly made plans to pursue a rematch. Earlier this year, the F.B.I. raided Mr. Cuellar’s Laredo home as part of an ongoing investigation that appears to be linked to an inquiry into the political influence of Azerbaijan, the former Soviet republic. In March, Mr. Cisneros fell less than 1,000 votes short of Mr. Cuellar’s total, forcing the two into the May 24 runoff.

Mr. Cuellar, who declined to comment for this article, has long defended his anti-abortion stance.

“As a lifelong Catholic, I have always been pro-life,” he said in a statement this week. “As a Catholic, I do not support abortion, however, we cannot have an outright ban. There must be exceptions in the case of rape, incest and danger to the life of the mother.”

Mr. Cuellar has repeatedly insisted that his stance mirrors the views of voters in the district. Though there has not been any public district-level polling on the issue, 2018 data from the Public Religion Research Institute found that Hispanics in Texas are less likely than other Americans to say they believe abortion should be legal in all cases, with 53 percent saying it should be illegal in most or all cases. Gallup’s Values and Beliefs poll last year found that low-income voters are also more likely to identify themselves as “pro life,” a trend that has held steady for several years and could be a factor in Mr. Cuellar’s largely working-class district. But in a poll from the Texas Politics Project at the University of Texas at Austin released last June, 54 percent of Hispanics said they were opposed to a ban on abortion if Roe v. Wade was overturned.

Just two days after the draft opinion was leaked, Mr. Cuellar campaigned alongside Representative James E. Clyburn, the House majority whip and third-ranking Democrat. Mr. Clyburn’s visit, which had been announced before the leaked opinion, put him in an awkward spotlight — supporting the only Democrat to vote against the Women’s Health Protection Act, a bill that would have codified Roe v. Wade into law.

Mr. Clyburn defended the incumbent, saying Democrats are a “big-tent party.”

“I don’t believe we ought to have a litmus test in the Democratic Party,” he told reporters in San Antonio. “We have to bring as many people into the party as we possibly can.”

Mr. Cuellar also has the support of Nancy Pelosi, the House Speaker, and Steny Hoyer, the majority leader. He frequently touts their endorsements, leaving many progressives privately grumbling that party leaders’ promises to fight for abortion rights in Congress ring hollow.

But several Texas Democratic officials cautioned against the idea that an intense focus on abortion would reshape the race. Asked how the leaked draft Supreme Court opinion overturning Roe v. Wade might change the runoff, Gilberto Hinojosa, the chairman of the Texas Democratic Party, sounded skeptical that it would fundamentally alter the dynamics of the contest, as some advocates have predicted. South Texas, he said, is “a different place.”

Angie Martinez, a 40-year-old bartender, said that while she supports abortion rights, it will not prevent her from voting for Mr. Cuellar.

“People are happy with the guy,” she said, adding that he had helped bring more funding into the district. “If abortion ends, it’s OK. Women go to Nuevo Laredo for abortions,” she said, referring to the Mexican city across the border from Laredo.

But Maxine Rebeles, a 39-year-old middle school writing teacher, said she is eager to see Mr. Cuellar out of office.

“He doesn’t protect our water, and he doesn’t protect our women,” Ms. Rebeles said. “When the government forces women to have children too young, bring children into unloving households, born to mothers who don’t love themselves, they get into a bad cycle.”

James Dobbins reported from Laredo, Jennifer Medina from Los Angeles and Katie Glueck from New York.