In a meeting with civil rights and abortion rights advocates on Monday, Vice President Kamala Harris said the administration plans to increase its focus on the issue in the final weeks of the campaign, according to someone briefed on the event who asked not to be named, discussing private conversations. Within hours of Mr. Graham’s introducing his proposal, Democrats across the country attempted to tie their opponents to the plan, saying that if elected, they would be “automatic votes” for a national ban in Congress.

“We can look at all the differences between where they have been, things that have been said, but at the end of the day what is being proposed is a national ban on abortion, and that stands in direct contrast with the will of the American people,” said Laphonza Butler, the head of Emily’s List, the largest funder of female Democratic candidates who support abortion rights.

For decades, abortion was an easy rallying call for the Republican Party. Politicians and party officials united with the active conservative base behind the broad, and somewhat vague, “pro-life” label.

But the court’s ruling forced politicians to confront the far messier specifics of abortion policy. In a debate in Indiana over the summer, Republican lawmakers viciously fought over how far a total ban should go, wrestling under the national spotlight with questions about child rape, ectopic pregnancy and life-threatening medical complications from pregnancy.

Some Republican political strategists are urging candidates to avoid talking about abortion as much as possible, saying that economic issues remain the number one concern for voters — particularly swing voters.

“My advice is, don’t take the bait. If you are pro-life, say you are pro-life,” said Kristin Davison, a G.O.P. strategist who worked on Gov. Glenn Youngkin’s upset win in Virginia last year. “Then, get back to the kitchen table.”