In Indiana, a law signed in 2016 by Mike Pence, then the governor, aims to ban discrimination against a fetus on things like race, sex, and disability. Though it has passed on the case before, the Supreme Court could take it up as soon as next week, and argue it next term. “We are hoping to challenge Roe from this angle, the angle of discrimination,” said Sue Liebel, the state director for the Susan B. Anthony List. “It has never been tried before.”

Even if the first primary challenge does not come from Indiana, the nationwide momentum is “really good news” for the anti-abortion movement, she said.

“It probably will not be one case that will topple Roe all at once,” Ms. Liebel said. “It will probably be multiple pieces that will take chunks out of Roe.”

In Ohio, the state legislature is prepared to approve a bill this session that would ban abortion after a fetal heartbeat is detected, which could be as early as six weeks into a pregnancy. Anti-abortion activist groups like Right to Life Ohio championed the bill, while abortion rights advocates have pointed out that many women and girls are not even aware that they may be pregnant that early.

The legislation was initially approved by the legislature last year, but was vetoed by John Kasich, then the governor. But his successor, Mike DeWine, who like Mr. Kasich is a Republican, has said that he intends to sign the legislation.

“We were very hesitant on the heartbeat bill because we knew we had a hostile Supreme Court,” Mike Gonidakis, the president of Right to Life Ohio, said about the court before the elevation of Justices Brett M. Kavanaugh and Neil Gorsuch by President Trump. “The time is ripe to have the discussion now because of the current Supreme Court. We now see a pathway forward.”