WASHINGTON — Republicans have spent decades attacking the landmark Supreme Court decision that legalized abortion nationwide, but with the toppling of Roe v. Wade seemingly imminent, their leaders in Congress and around the country have grown suddenly quiet on the issue, part of a bid to avoid a backlash against their party ahead of the midterm elections.

In the days after the leak of a draft Supreme Court opinion that would overturn the 50-year-old precedent, Republicans in Congress have notably refrained from taking a victory lap for having helped to install the conservative majority that has paved the way for such an outcome.

Even as some of their counterparts at the state level race forward with far-reaching abortion bans that could even affect some methods of contraception, Republicans appear determined to recast their position on the issue as one of moderation and avert the gaze of voters away from their anti-abortion-rights agenda.

“You need — it seems to me, excuse the lecture — to concentrate on what the news is today,” Senator Mitch McConnell of Kentucky, the Republican leader, said on Tuesday. “Not a leaked draft but the fact that the draft was leaked.”

The Republicans’ caution reflects the potential for the eventual ruling to change the midterm political landscape. Their leaders and candidates have built a campaign to reclaim control of the House and Senate around inflation, economic uncertainty, crime, border control and American doubts that President Biden, who is deeply unpopular, can right the ship.

Now the prospect of eliminating abortion rights has added a tectonic change to American life into the mix, threatening to upend that focus.

Democrats have signaled that they plan to use the coming decision as a rallying cry for voters to reject Republicans, portraying its implications as vast and unacceptable.

“This is an issue that is defining for this country today, and if the American people don’t stand up for equality for every American at this moment in time, we will be undermining a right to privacy in more than this context,” said Senator Kirsten Gillibrand, Democrat of New York. She raised the specter of a conservative Supreme Court going after gay marriage, consensual same-sex relations and even contraception if the decision stands.

Republicans, by contrast, believe their candidates’ job right now is to remain focused on the economy and not allow any other issue — particularly one that could alienate suburban independent voters whose backing they need to win congressional majorities — to distract them.

“Big picture, tell me what the 30-year fixed mortgage rate will be and if anything has improved with gas and groceries, and I’ll tell you the results,” said Corry Bliss, a veteran strategist who advises Republican candidates. “That is what the midterms are going to be about — period, end of discussion.”

Republicans are talking about abortion, just not openly. A document circulated by the National Republican Senatorial Committee and obtained by Axios urged candidates to be low-key about the issue, with a post-Roe America looming as early as next month.

“Abortion should be avoided as much as possible,” the document advised candidates to say. “States should have the flexibility to implement reasonable restrictions.”

Republicans do not want to throw doctors and women in jail, the document continued. They certainly do not want to take away contraception. And if any party is being extreme, it instructed Republicans to argue, it is the Democrats, who will not accept even modest restrictions on abortion that most Americans support.

The approach is calculated to exploit the fact that Democrats, outraged about the ruling yet powerless to do anything about it, are planning a symbolic vote that puts their party on the record opposing almost any abortion limits. On Wednesday, Senate Democrats will try — and likely fail — to take up legislation that would not only codify the right to an abortion, but also nullify restrictions that have passed muster with the courts.

“The Democrats are going to make this easy for us,” said Mallory Carroll, vice president of communications at Susan B. Anthony List, which works to elect officials who oppose abortion rights. She called the Democrats’ Women’s Health Protection Act “far outside the American mainstream.”

And “mainstream” is how the Republican campaign arms want their candidates to present themselves — as soft-spoken, compassionate, “consensus builders,” as the talking points put it.

“I am pro-life, but this isn’t about political labels,” the documents suggest Republican candidates say. “I believe all Americans want us to welcome every child into the world with open arms. But if you disagree with me, my door’s always open.”

Governors like Brian Kemp of Georgia and Ron DeSantis of Florida have said relatively little on the issue since the draft opinion came out.

Even former President Donald J. Trump, who campaigned in 2016 on appointing Supreme Court justices who would overturn Roe, has refrained from gloating.

“Nobody knows exactly what it represents,” he told Politico, calling the leak of the opinion “a terrible thing for the court and for the country.

“We’ll talk about it after we find out what the definitive version is,” he said.

It is still possible that the court will not go as far as the draft. Chief Justice John Roberts confirmed that the leak was authentic but cautioned that the decision was not final.

Still, the problem for Republican leaders in Washington who want to downplay the implications of the potential ruling is the very clear message coming from their party’s state legislators about the severe restrictions many would enact if there were no longer a right to an abortion in the Constitution.

On Wednesday, lawmakers in Louisiana pushed forward legislation that would do precisely what the Washington talking points deny: grant constitutional rights to “all unborn children from the moment of fertilization,” and classify abortion as homicide. Such a law could, in fact, put women and doctors in prison and ban certain types of contraception, such as IUDs or morning-after pills, that block implantation of a fertilized egg.

Republicans in South Dakota, Indiana and Nebraska have called for special sessions of their legislatures to move on strict abortion bans as soon as a final decision is announced.

And for all the caution Republican strategists might advise, there is still the passion of the issue. New Hampshire State Representative Susan DeLemus was filmed responding to abortion-rights protesters at the State Capitol in Concord, N.H., on Thursday by screaming that they were “murderers.”

To a certain extent, opponents of abortion say they really are in a moment of unreality. Carol Tobias, president of the National Right to Life Committee, said she had been preparing for a good night’s sleep in her Albuquerque home when the news first broke of the draft opinion. She said she thought someone had pulled a prank on the reporters.

Even now, she said she is cautious.

“What came out earlier this week is three months old,” she said. “I certainly hope it is the final draft, but I’ve been told it isn’t. It’s still possible a justice or two has changed positions.”

Republicans say their restraint on the issue makes sense. A near-total abortion ban has been in place in Texas for eight months, and seemingly no political price has been paid so far.

State Representative Gene Wu, a Houston Democrat, said he worries that in Republican states that have been living with steadily rising restrictions on reproductive rights, the response to overturning Roe could be as muted as it has been in Texas, and that in Democratic states, voters will be reassured that their rights are safe.

“This has been done so incrementally, it’s like there’s a learned helplessness. We’ve taken so much abuse; what’s a little more?” he said, likening women in states like Texas to the frog in the boiling pot of water. “I hope that’s not the case.”

Another factor mitigating the backlash might be the rising popularity of long-term contraception, such as IUDs, and the increased access to birth control in general, which has helped lower the nation’s abortion rate in recent years and given more women a sense of reproductive security.

A decade-old study by the American College of Obstetricians and Gynecologists found that the percentage of women in childbearing years using long-acting, reversible contraception had risen steadily, from 2.4 percent in 2002 to 8.5 percent in 2009 to 11.6 percent in 2012. The figure is about 12 percent now, said Dr. Nisha Verma, a fellow at the college and a gynecologist in Washington, D.C.

“The need for abortion will never go away,” Dr. Verma said, but, she added, “We’ve definitely seen that people have been able to take more control in their reproductive health.”

Another study in the Journal of the American Medical Association found a 21.6 percent jump in the use of such contraception in the months after the 2016 election of Mr. Trump, with his vows to install justices who would overturn Roe.

Emily Cochrane contributed reporting.