MONTGOMERY, Ala. >> About 20 women came to the abortion clinic here on a recent morning, hurrying past the shouting protesters as volunteer escorts held up umbrellas to shield their faces.

Inside the Reproductive Health Services clinic was Dr. Willie Parker, an Alabama native and one of a few physicians willing to face the professional shunning and the personal threats that come with being an abortion doctor in the conservative Deep South. He travels constantly among three cities, two in Alabama and one in Mississippi, to provide a service that no local doctors will.

Despite being an experienced, board-certified physician, Parker, 53, said he had been unable to get the admitting privileges to local hospitals that Alabama and Mississippi have tried to require of abortion doctors. Because federal courts have temporarily blocked those requirements here and across the state line in Mississippi, Parker continues to practice, and the clinics that rely on him are still open.

But the future of this clinic and many others, across the South and much of the country, could be at stake this spring as the Supreme Court takes up what both sides in the abortion debate describe as a landmark case. While the death of Justice Antonin Scalia has added new considerations, the court’s decision in the case, which involves a Texas law, could shape abortion rules for years to come.

Highlighting the wider stakes, Wednesday the 5th U.S. Circuit Court of Appeals, overruling a lower court, said a Texas-style admitting-privileges law in Louisiana that is currently blocked should take immediate effect — forcing three of the state’s four abortion clinics to close. Lawyers for the clinics were planning an emergency appeal to the Supreme Court on Wednesday night, arguing that the law should continue to be blocked while the justices consider whether such laws are constitutional.

On March 2, the Supreme Court is scheduled to hear arguments on the challenge to the Texas law, which requires abortion doctors to be affiliated with nearby hospitals and also limits abortion to ambulatory surgical centers. Abortion opponents say such measures are needed to protect women, but major medical groups say they will not enhance patient safety and will only reduce women’s access to abortion.

Overruling a lower court’s injunction, the 5th Circuit allowed the Texas admitting-privilege rule to take effect throughout the state in 2013, immediately shuttering about half of what had been more than 40 abortion clinics, although exceptions were later granted for geographically isolated clinics in McAllen and El Paso. The second requirement, mandating costly surgical center facilities, has been temporarily stayed by the Supreme Court, but it would force still more reductions if upheld.

At stake in the case, Whole Woman’s Health v. Hellerstedt, is not only the future of abortion access in Texas and in the nine other states that, like Alabama and Louisiana, have adopted similar physician rules. It could also affect dozens of other regulations of disputed medical value that have been adopted by numerous states, including limits on nonsurgical drug-induced abortions, mandated building standards for clinics and two-day or three-day waiting periods.

“The Whole Woman’s Health case will be a turning point in the battle over access to abortion services,” said Nancy Northup, president of the Center for Reproductive Rights in New York, which will be arguing before the Supreme Court on behalf of Texas abortion clinics and is filing the emergency appeal on behalf of Louisiana clinics.

“The decision is likely to have an effect on a whole range of laws that pretend to be about women’s health but actually are designed to close clinics,” Northup said.

Mike Gonidakis, president of Ohio Right to Life, which has successfully lobbied for numerous abortion regulations that an adverse Supreme Court decision could imperil, agreed on the stakes, if not the description.

“Everyone’s holding their collective breaths,” he said. “We believe this case is either going to expand the ability of states to regulate abortion, or limit it.”

At issue is the practical meaning of the Supreme Court’s landmark ruling in a 1992 case, Planned Parenthood v. Casey, that states may not impose an “undue burden” on access to abortion: A law is invalid “if its purpose or effect is to place substantial obstacles in the path of a woman seeking an abortion before the fetus attains viability.” Can the courts second-guess a legislature’s assertion that a rule promotes women’s health? Can a state enforce laws that seriously reduce access to abortion without valid medical reasons?

The timing of an ultimate resolution could be altered by the death of Scalia, but not necessarily. If the likely swing voter, Justice Anthony M. Kennedy, sides with Texas and the court splits 4-4, then the existing ruling by the 5th Circuit, permitting the law, would stand.

But a tie vote would not be a binding precedent, leaving uncertainty for other states and highlighting more than ever the importance of the next Supreme Court appointment. If the four anti-abortion votes were secure and the post was filled by another conservative, the court would quickly have opportunities to reach a binding decision next year, as states like Alabama, Mississippi and Wisconsin press to remove blocks on their admitting-privilege laws.

On the other hand, if the court strikes down the Texas law by a 5-3 vote, Scalia’s absence would have no effect and even the later appointment of a conservative would not alter the balance. Such a majority decision could create a bulwark for abortion clinics.

In a sign that many more Republican-led states could adopt admitting-privilege and similar restrictions if the Supreme Court gives a green light, 23 states, including Alabama this month, have joined in a brief urging the court to uphold the Texas restrictions. They argue that health and safety rules should be left to the states.

Here in Alabama, a binding Supreme Court ruling that upholds admitting-privilege rules would have sweeping consequences.

Right now, only one doctor, at a clinic in Huntsville in the far north of the state, has local hospital privileges. But if the law is allowed to take effect, four of the state’s five clinics would close, and the lone surviving clinic could never meet the demand for abortions in Alabama, which average around 9,000 a year, said June Ayers, owner and director of Reproductive Health Services.

Declaring Alabama’s law unconstitutional in August 2014, Judge Myron H. Thompson of U.S. District Court in Montgomery rejected the state’s argument that admitting privileges would enhance patient care. If this clinic-closing requirement did not “constitute an impermissible undue burden,” he wrote, “then almost no regulation, short of those imposing an outright prohibition on abortion, would.”

Condemning the decision and vowing to appeal, Gov. Robert Bentley reiterated his opposition to abortion as he stated: “As a doctor, I firmly believe that medical procedures, including abortions, performed in Alabama should be done in the safest manner possible. This law ensures that if a complication arises there is continuity of treatment between doctor and patient.”

The Montgomery clinic juggles the schedules of three traveling doctors, including Parker.

Already, many of the 1,000 to 1,200 women obtaining abortions at this clinic each year face hours of driving, Ayers said, and all must make the trip twice because the state requires a 48-hour waiting period after the first visit, which abortion opponents hope will cause those planning to end their pregnancies to have second thoughts. More than two-thirds of the clinic’s patients live at or below the poverty line, and a large majority already have at least one child, she said.

Ashley Garza, 29, who retired after six years in the Air Force that included service in Afghanistan, was in the clinic for an abortion the other day. She and her boyfriend, who plan to eventually marry and have a family, had driven two hours from the southeastern corner of Alabama.

“If I had a child now, we’d be in absolute poverty,” said Garza, who scrapes by on the GI Bill while pursuing a degree in social work and who described her own childhood as one of extreme hardship. “It wouldn’t be fair to the child.”

From their town, it would be a five-hour drive to the sole surviving clinic, in Huntsville, which, Garza said, “a lot of women just couldn’t do.”