The lines have hardened in response to the Supreme Court’s rightward turn. President Trump’s appointment of conservative Justice Brett Kavanaugh last year solidified a conservative majority on the high court. Many players on both sides say that raises the chances the justices could eliminate abortion rights or at least give states more freedom to enact stricter standards and restrictions.

“All the states are looking to the Supreme Court,” said Elizabeth Nash, a policy expert at the Guttmacher Institute, a research group supportive of abortion rights. “Adding Kavanaugh to the court has totally reshaped the debate at the state level.”

Whether the new lineup of justices will produce a major shift on abortion remains unclear. On Thursday, Chief Justice John Roberts and the court’s liberal bloc voted to suspend enforcement of a physician-credentialing regulation in Louisiana similar in language to a Texas statute struck down as unconstitutional in 2016. The 5-4 ruling put the law on hold while the high court considers an appeal of a lawsuit challenging it.

Abortion is among a host of national issues polarizing state politics, along with immigration enforcement, voter-registration requirements, religious rights and gun control. The fault lines have widened as fewer states have divided governments. The same political party occupies the governorship and controls both legislative chambers in 36 states, the highest number in decades, according to Ballotpedia. “You’ve got trains on parallel tracks heading in opposite directions,” said Karl Kurtz, a state legislative scholar.

In conservative states, lawmakers are pushing for “fetal heartbeat” bills that would ban abortion after detection of a heartbeat, which can be as early as six weeks into a pregnancy.

A federal appeals court in 2015 struck down heartbeat bills enacted in North Dakota and Arkansas, citing the 1973 Roe v. Wade ruling that established a woman’s constitutional right to terminate a pregnancy before viability.

More recently, an Iowa judge in January blocked enforcement of a heartbeat law signed last year by Republican Gov. Kim Reynolds.

Despite the judicial setbacks, Republican governors in at least four states — Ohio, Tennessee, Florida and South Carolina — have signaled support for heartbeat bans.

The state’s new GOP governor, Mike DeWine, has said he would sign a heartbeat ban, unlike his predecessor, John Kasich, who vetoed similar legislation in 2016. “Ultimately, this will work its way up to the United States Supreme Court,” Mr. DeWine told a conservative radio host last month.

Lawmakers in Kentucky, where Republicans control the legislature and the governorship, also have introduced a heartbeat bill, drawing objections from Attorney General Andy Beshear. A Democratic candidate for governor, Mr. Beshear warned legislative leaders in a memo that the legislation was unconstitutional and would trigger a costly, unwinnable court fight.

Kentucky lawmakers aren’t sure they’ll pass the measure but say the issue is galvanizing their constituents. “The life issue isn’t just an undercurrent but a major, driving factor in elections,” Kentucky Senate majority floor leader Damon Thayer said.

Elsewhere, Utah lawmakers are looking at banning most abortions after 15 weeks. And more than 20 Kansas House lawmakers have backed a proposed constitutional amendment granting “inalienable rights” to fetuses from the moment of fertilization.

Liberal lawmakers in statehouses controlled by Democrats, meanwhile, are trying to safeguard and expand abortion protections.

National abortion-rights activists say if they can’t block the sweeping bans proposed by the Republicans, they want to shore up rights in liberal states with laws guaranteeing abortion access and, in some cases, expanding abortion access beyond the holdings of Roe v. Wade.

The movement’s goal is to “advance laws that will ensure reproductive rights will be protected in as many places as possible,” said Andrea Miller, president of the National Institute for Reproductive Health.

Last month, abortion-rights activists cheered a law signed by New York Democratic Gov. Andrew Cuomo that elevated abortion to a “fundamental right,” repealed abortion offenses from the criminal code and made it possible for non-doctors such as trained nurses to perform procedures. A new health-code provision says doctors may perform third-trimester abortions of viable fetuses but only if necessary to protect the life or health of a mother—a broadly interpreted exception based on Supreme Court guidance—although it doesn’t specify penalties for violations.

Republicans, including President Trump in his State of the Union address Tuesday, have blasted New York’s law as a license to kill babies. Several Roman Catholic bishops called for the excommunication of Mr. Cuomo, who is Catholic. The governor and Democrats say the criticisms are grossly misleading, hailing the abortion protections as a leap forward in women’s rights.

In Virginia, Democrats tried to advance similar legislation but retreated after facing blowback from a viral video of the bill’s sponsor discussing how it could in rare circumstances allow a woman in labor to get an abortion to protect her mental health.

Vermont is considering a bill making abortion a “fundamental right” and establishing that a “fetus shall not have independent rights” under state law.

Massachusetts lawmakers have proposed eliminating parental consent requirements for minors. “In this uncertain environment, we in Massachusetts have an opportunity —and an imperative—to be bold,” wrote NARAL Pro-Choice Massachusetts on its website. The bill “goes above and beyond the protections of Roe v. Wade,” the group said.

And in New Mexico, where Democrats control both houses and the governorship, lawmakers are sparring over whether a Democratic-backed bill formally decriminalizing abortion would preserve the right of doctors to refuse to perform abortions on moral or religious grounds.