Under the hot desert sun less than 40 miles from the U.S.-Mexico border last week, Representative Gabe Vasquez, Democrat of New Mexico, lashed out against Republicans who talk about abortion as a “states’ rights” issue, charging that their real agenda is to place severe limits on women’s access to health care.

“That is now the de facto position of folks who want to limit women’s reproductive health care,” Mr. Vasquez told about 50 people at a groundbreaking for a new, $10 million state-funded reproductive health clinic. “We’ve seen the consequences of what this means for women.”

It is a message that Mr. Vasquez, a first-term congressman in a highly competitive House district, has turned to time and again as he fights to keep his seat amid a challenge from former Representative Yvette Herrell, the Republican he defeated two years ago.

He has worked to use Ms. Herrell’s anti-abortion rights voting record and previous remarks on the issue against her, including a 2020 video clip in which she said during a Republican candidate forum: “I wish we could have eliminated all abortion in this state.”

Ms. Herrell, unlike some Republicans who have airbrushed or downright misrepresented their past records on the issue, says she is proud of her stance on abortion, including her backing for legislation that would grant legal personhood to fertilized eggs, effectively criminalizing the termination of a pregnancy and potentially aspects of in vitro fertilization treatment.