The justices may yet consider whether the 2014 law - requiring doctors at abortion clinics to have admitting privileges at nearby hospitals - unduly burdens a woman's access to abortion. The Louisiana law has never been enforced, and the Supreme Court in 2016 found a nearly identical Texas law to be unconstitutional.
"The Supreme Court has stepped in under the wire to protect the rights of Louisiana women," said Nancy Northup, president and CEO of the Center for Reproductive Rights, which represented the law's challengers.
"The three clinics left in Louisiana can stay open while we ask the Supreme Court to hear our case. This should be an easy case - all that's needed is a straightforward application of the court's own precedent."
The court's four most conservative members would have allowed the law to take effect. Justice Brett Kavanaugh said there was a dispute about whether the doctors could obtain admitting privileges, and that a 45-day grace period would have given time to settle that question.
"The parties have offered, in essence, competing predictions" about whether several doctors can obtain privileges, Kavanaugh wrote.
"If we denied the stay, that question could be readily and quickly answered without disturbing the status quo or causing harm to the parties or the affected women, and without this court's further involvement."
Justices Clarence Thomas, Samuel Alito Jr. and Neil Gorsuch would have allowed the law to go into effect, but they did not join Kavanaugh's dissent.
But the abortion providers in their briefs had said just the prospect that the law would go into effect was already affecting services.
"Scheduled medical procedures are being cancelled, physicians and clinic staff are preparing to be out of work, and patients seeking to exercise their constitutional right to abortion are being turned away or sent to other states," their brief stated.
The majority, as is custom, did not give a reason for granting the stay. But it seems likely the full court will now grant the case a full briefing and review, and perhaps reexamine its earlier decision, which was made by a very different Supreme Court.
In the court's 2016 decision, it said the admitting privileges requirement "provides few, if any, health benefits for women, poses a substantial obstacle to women seeking abortions, and constitutes an 'undue burden' on their constitutional right to do so."
Hospitalization after an abortion is rare, all sides agree, and the lack of admitting privileges by the doctor who performed the procedure is not a bar to the woman getting needed medical care.
But last fall, a panel of the U.S. Court of Appeals for the 5th Circuit upheld the Louisiana law in a 2-to-1 vote, finding factual distinctions between how the restriction played out in Texas and Louisiana. The full court, considered one of the most conservative of the regional appeals courts, voted not to reconsider the panel's decision.
Dissenting judges said their colleagues in the majority ignored requirements set out in the Supreme Court's 2016 decision and seemed intent on giving the high court reason to reconsider that precedent, called Whole Woman's Health v. Hellerstedt.
The Whole Woman's Health decision was the court's most important one on abortion in a quarter-century. But it was decided by a very different eight-member Supreme Court in the wake of conservative Justice Antonin Scalia's death.
Justice Anthony Kennedy joined the court's liberals to strike down the Texas provisions. Roberts, Thomas and Alito dissented.
Those three have been joined by President Donald Trump's choices, Gorsuch and Kavanaugh, both of whom were supported by antiabortion activists who said they hoped that the court's new conservative majority might one day overturn the fundamental right to abortion the court advanced in Roe v. Wade.
The doctors and clinics who challenged the Louisiana law said allowing it to go into effect would provide a way for states to undermine the right to abortion without overturning Roe.
At stake "is not just the constitutional rights of Louisiana women to abortion access," wrote Julie Rikelman and Travis Tu of the Center for Reproductive Rights.
"The Fifth Circuit panel majority's decision undermines the rule of law by flouting binding precedent from this Court. Such a ruling has implications for the country and the judicial system as a whole."
Judge Jerry Smith, writing for the two-member appeals court majority, said that the court complied with the Supreme Court's decision in Whole Woman's Health by taking a painstakingly close look at the details.
"Unlike in Texas, the [Louisiana law] does not impose a substantial burden on a large fraction of women," he concluded.
He said the closing of some clinics in Louisiana, as opposed to Texas, would not dramatically increase driving distances, and that it was easier for doctors in Louisiana to obtainadmitting privileges. The "vast majority" of the six doctors who performed abortions in Louisiana "largely sat on their hands" instead of working hard to procure the credential, Smith wrote.
He concluded that "at most, only 30 percent of women" seeking abortions in Louisiana would be affected.
"The record here indicates that the admitting-privileges requirement performs a real, and previously unaddressed, credentialing function that promotes the wellbeing of women seeking abortion," Smith wrote.
Still, he acknowledged, the benefits are "not huge." In a footnote, Smith wrote that "the state did not provide any instance in which a worse result occurred because the patient's abortion doctor did not possess admitting privileges."
Dissenting judges and the challengers said it was a fundamental mistake to approve a law that imposes any burden on a woman's right to an abortion when it provides no corresponding benefit for the woman's health.
Hospitalization occurs in only 0.05 percent of abortions in the first trimester and approximately 1 percent in the second trimester, the challengers said. Hospital care, when needed, is provided regardless of whether the doctor performing the abortion has admitting privileges.
The appeals court rejected a district judge's finding that the law would affect about 70 percent of women seeking abortions in the state. After a trial, that judge found that two of the remaining three abortion clinics in the state would have to close because they would not have a doctor who could obtain admitting privileges.
The only remaining clinic with an eligible doctor would be in New Orleans, the district court said, and incapable of meeting the demand of approximately 10,000 abortions.
(COMMENT, BELOW)