Democrats seized on a ruling on Tuesday by Arizona’s highest court upholding an 1864 law that bans nearly all abortions, setting up a fierce political fight over the issue that is likely to dominate the presidential election and a pivotal Senate race in a crucial battleground state.

Even though the court put its ruling on hold for now, President Biden and his campaign moved quickly to blame former President Donald J. Trump for the loss of abortion rights, noting that he has taken credit for appointing the Supreme Court justices who overturned a constitutional right to abortion. Just a day earlier, Mr. Trump had sought to defang what has become a toxic issue for Republicans by saying that abortion restrictions should be decided by the states and their voters.

Mr. Trump offered no immediate response to the decision, and his aides did not respond to requests for comment.

Nowhere are the politics of abortion more distilled than in Arizona, where liberal advocates have been pushing for a ballot measure in November that would enshrine abortion rights in the State Constitution. Supporters of the measure say they have already gathered enough signatures to put the question on the ballot ahead of a deadline in early July.

That means the state is likely to be front and center in a national push by Democrats to transform the 2024 race into another referendum on abortion rights.

The issue has emerged as one of the party’s strongest political weapons since the Supreme Court decision to overturn Roe v. Wade in 2022, powering them to a series of electoral victories. The Arizona ruling on Tuesday will pose yet another test for Republicans, who after decades of efforts to limit abortion rights and access have struggled to find a winning message on the issue amid the transformed politics of the post-Roe era.

The 1864 law will not be enforced immediately: In putting its ruling on hold, the court sent the matter back to a lower court to hear additional arguments about the legislation’s constitutionality.

Still, Democrats quickly aimed to capitalize on the news.

“This ruling is a result of the extreme agenda of Republican elected officials who are committed to ripping away women’s freedom,” President Biden said in a statement minutes after the decision.

Vice President Kamala Harris was also set to travel on Friday to Tucson, Ariz., to talk about the importance of abortion rights. She has become a leading messenger for the Biden campaign on abortion, becoming the highest-ranking U.S. official to visit an abortion clinic. She has also held events on abortion in key battleground states including Wisconsin, Georgia and Michigan.

On Tuesday, she squarely blamed Mr. Trump for the court’s decision — a tactic Democrats have taken on abortion nationwide.

“Arizona just rolled back the clock to a time before women could vote — and, by his own admission, there’s one person responsible: Donald Trump,” she said in a statement.

The condemnation of the ruling was bipartisan, reflecting the continued power of the issue since the overturning of Roe in 2022. Both candidates in Arizona’s Senate race, which is seen as one of the most important contests for control of the chamber, quickly disavowed the ruling.

Representative Ruben Gallego, the Democratic candidate, called it “devastating for Arizona women and their families” and warned that “women could die” as a result of the new ban.

More strikingly, Kari Lake, his Republican opponent, echoed his comments, despite having praised the law in the past.

“It is abundantly clear that the pre-statehood law is out of step with Arizonans,” she said in a statement, adding, “This is a very personal issue that should be determined by each individual state and her people.”

Democrats quickly blasted out examples of Ms. Lake’s past support for the 19th-century legislation, including comments in which she called it a “great law.”

Polling has shown that Mr. Biden has a clear edge over Mr. Trump on abortion, despite voters’ preference for the former president on almost every other major issue. The political calculus for Mr. Biden and other Democrats is that every day in which abortion is the primary political topic is a better day than one spent discussing the economy, immigration or thorny foreign policy issues.

Since the fall of Roe, Democrats have repeatedly won elections by making abortion the main issue in their campaigns. Last year, a liberal Wisconsin judge won a commanding victory in the state’s crucial Supreme Court race, and Gov. Andy Beshear of Kentucky won re-election in a deep-red state — both by focusing heavily on abortion rights.

The Biden campaign believes abortion rights are one of the strongest issues working in its favor in Arizona, where independents make up about a third of the electorate. In March, 50 percent of registered voters in the state said they trusted Mr. Biden to do a “better job” on abortion, compared with 44 percent for Mr. Trump, a Fox News poll found.

“We have folks across the spectrum who are going to be outraged by this decision and who are going to hold the person who brags about being responsible for it to account,” said Jen Cox, a senior adviser for the Biden campaign in Arizona. “And that’s Donald Trump.”

Abortion providers said they expected to continue performing abortions through May as their lawyers and Democratic lawmakers search for new legal arguments and additional tactics to delay the ruling.

If reinstated, the law would pre-empt the state’s current restriction on abortion after 15 weeks with a total ban outlawing the procedure from the moment of conception, except when necessary to save the life of the mother. The 1864 law contains no exceptions for rape or incest. Doctors prosecuted under the law could face fines and prison terms of two to five years.

Leaders of the anti-abortion movement celebrated the ruling as a major step forward.

Marjorie Dannenfelser, the president of Susan B. Anthony Pro-Life America, a prominent anti-abortion group, praised the ruling as an “enormous victory for unborn children and their mothers,” saying her movement “must continue to fight.”

But they largely stood alone in their support for the ruling. Some Arizona Republicans who are facing competitive re-election fights criticized the decision, despite their past opposition to abortion rights.

Representative Juan Ciscomani, who said in 2022 that abortion law should be left to the states, called the ruling a “disaster for women and providers.” And Representative David Schweikert, who previously expressed support for the overturning of Roe and for abortion bans, wrote on social media, “This issue should be decided by Arizonans, not legislated from the bench.”

Still, the Arizona ruling underscored the political limitations of efforts by Republicans to skirt specific questions about the future of abortion access in more conservative states.

In Mr. Trump’s remarks on Monday, he supported exceptions for rape, incest and life of the mother, but he did not offer an opinion on whether state bans that do not include those caveats — like the one in Arizona — should stand.

Reporting was contributed by Ruth Igielnik, Jack Healy, Annie Karni, Kellen Browning and Michael Gold.