A leaked draft of a Supreme Court ruling overturning Roe v. Wade instantly propelled the debate over abortion into the white-hot center of American politics, emboldening Republicans across the country and leaving Democrats scrambling to jolt their voters into action six months before the midterm elections.

Although the Supreme Court on Tuesday stressed that the draft opinion was not final, the prospect that the nation’s highest court was on the cusp of invalidating the constitutional right to abortion was a crowning moment for Republicans who are already enjoying momentum in the fight for control of Congress, statehouses and governor’s offices. Republican state leaders on Tuesday announced plans to further tighten restrictions on the procedure — or outlaw it outright — once the final ruling lands in the coming months.

Democrats, reeling from the blow and divided over whom to blame, hoped the news would serve as a painful reality check for voters who have often taken abortion rights for granted and struggled to mobilize on the issue with the passion of abortion rights opponents. They said they planned to drive home the stakes in the fall, particularly in state races, putting abortion rights on the November ballot in key contests in Pennsylvania, Michigan, Arizona and other battlegrounds.

“People were concerned about the lack of energy for voters in the midterms and not coming out to vote — well, the Supreme Court has just handed us a reason for people to vote,” said Representative Susan Wild, a Pennsylvania Democrat who faces a competitive re-election.

“At one time I would have said they’re never going to take away right to contraception. But I don’t believe that anymore,” she said.

Independent voters have overwhelmingly soured on President Biden, and many core Democratic constituencies have shown signs of trouble. Some party strategists privately cautioned against the idea that even something as seismic as overturning Roe would surpass the importance of the economy and inflation with many voters, something Republicans argued publicly.

“Conventional wisdom right now is this helps Democrats because it will spur turnout, but it also could certainly spur turnout for base Republicans,” said Glen Bolger, a Republican strategist. “Generally most voters focus on the economy, for instance, and right now of course, inflation is dominant.”

But polling also shows that Americans strongly oppose completely overturning Roe v. Wade — 54 percent of Americans think the Roe decision should be upheld while 28 percent believe it should be overturned, a new Washington Post-ABC poll found. Democrats argue that many voters have long believed it was not truly in danger of being gutted. The draft opinion may change their calculus in meaningful ways, especially with suburban women and disillusioned base voters, those strategists say.

“It hasn’t ever been that voters don’t care about it,” said Molly Murphy, a Democratic pollster and strategist, and the president of Impact Research. “It’s been concluded that it’s less effective because voters don’t believe that it could actually go away. And so with what the Supreme Court is signaling they’re about to do, is completely change and eliminate that sort of theory of the mobilizing power of abortion.”

Without the court’s protection for abortion rights, states would be free to enforce their own restrictions or protections. That patchwork system is likely to shift the focus to governor’s races, where a state’s executive could have an outsize role in determining whether abortion is legal.

In Pennsylvania, Josh Shapiro, the state attorney general and Democratic candidate for governor, signaled that he planned to seize on the looming threat to Roe to cast himself as a one-man firewall against abortion rights opponents in his state. On Tuesday, he pledged to veto any legislation from the Republican-controlled Pennsylvania legislature that would restrict abortion access.

“Every Pennsylvanian should be able to raise a family on their own terms,” Mr. Shapiro said. “And that means deciding if and when and how they want to do that.”

But for all the talk from Democrats about abortion being on the ballot this fall, Mr. Shapiro’s race is the exception. Far more states, including Arizona, Georgia, Michigan, Texas and Wisconsin, all have laws on the books effectively banning abortion that would go into effect once Roe is invalidated. The November elections are unlikely to give Democrats the numbers to reverse those.

In Wisconsin, for example, an 1849 law made performing an abortion a felony unless the pregnancy endangered the life of the mother. That law remains on the books, though several of the state’s Republican candidates for governor have endorsed proposals to eliminate any exceptions to the ban.

On Tuesday afternoon, Gov. Tony Evers of Wisconsin sent a letter, signed by 15 fellow Democratic governors, urging Congress to enact federal abortion protections — a plea that is almost certain to go unmet.

Although Mr. Evers won’t be able to make the case that he can save abortion protections in Wisconsin, he will argue that he can make other key decisions about how much the machinery of the state is used toward investigations and prosecutions of abortions, said Ben Wikler, the chairman of the Democratic Party of Wisconsin.

Republicans were celebrating as they appeared on the cusp of victory. In Georgia, Gov. Brian Kemp, a Republican facing stiff primary and general election challenges, took a victory lap Tuesday, playing up a 2019 state law that bans abortion in the state after six weeks. The law has been held up in a federal appeals court awaiting the outcome of the Supreme Court’s decision.

“We are the voice of all those people that are out there and have been in the trenches for decades doing this and we’re glad to be in the fight with them,” Mr. Kemp said during a radio interview Tuesday.

In South Dakota, Gov. Kristi Noem, a Republican believed to have presidential ambitions, said Tuesday that she would immediately call for a special session to outlaw abortion in her state. Attorney General Eric Schmitt of Missouri said a broad ban on abortions in the state was just a signature away from enactment if Roe is in fact overturned. The speaker of the Nebraska Legislature told colleagues to expect a special session on abortion following the Supreme Court’s decision.

Democrats running for Senate renewed calls to put Roe’s abortion protections into federal law and change the Senate rules, if necessary, to do it. Although Democrats currently control the Senate with Vice President Kamala Harris’s tiebreaking vote, they do not appear to have the votes to codify a woman’s right to an abortion, a major point of contention and blame-shifting among Democrats.

Senator Joe Manchin III of West Virginia said Tuesday that he was still opposed to any changes to the filibuster, effectively ending any Democratic hopes of passing an abortion bill.

Still, Democratic candidates signaled they planned to continue to promise to fight to codify Roe.

“Democrats have to act quickly and get rid of the filibuster,” said Lt. Gov. John Fetterman of Pennsylvania, who is running for Senate, to “finally codify Roe into law. We cannot afford to wait.”

Kina Collins, a Democrat in a primary for a House seat in Chicago, called on the party’s leaders to “fight like our lives depend on it.”

“There is no place in this party for Democrats who will not,” she said.

Sensing the potential harm of yet another intraparty skirmish, Representative Sean Patrick Maloney of New York, the chairman of the House Democrats’ campaign arm, warned against blaming fellow Democrats.

“Focusing on what’s wrong with Democrats in the Senate or elsewhere is (another) circular firing squad,” Mr. Maloney wrote on Twitter. “We can only end the filibuster, pass real protections for choice IF WE WIN more power.”

Trip Gabriel contributed reporting.