Washington (CNN) The Supreme Court appeared divided after over an hour of arguments Wednesday concerning a controversial Louisiana abortion access law that critics say will leave just one doctor in the state to perform the procedure.

Key justices indicated that they could be poised to uphold the law that is similar to a Texas law the Court struck down just four years ago. Chief Justice John Roberts and Justice Brett Kavanaugh both asked questions suggesting that the Louisiana law that requires doctors to have admitting privileges in local hospitals could serve a valid purpose.

But Roberts' vote could be essential, and at the end of oral arguments it wasn't completely clear where he would come out.

The case marks the first time President Donald Trump's nominees -- Neil Gorsuch and Kavanaugh -- have heard an abortion-related dispute at the high court. Supporters of abortion rights fear the Court's solid conservative majority is poised to cut back or even gut court precedent . It comes as Republican-led states are increasingly emboldened to pass laws that mark a potential rollback of abortion rights.

Justice Samuel Alito emerged as the strongest and most vocal supporter of the law and even questioned whether the doctors and clinics could actually bring the challenge because they might have a "conflict of interest" with the women seeking abortion. The doctors' interests in new regulations, Alito suggested, may not align with a patient seeking treatment.

Gorsuch and Justice Clarence Thomas said nothing.

While Louisiana argues the law will increase public safety, supporters of abortion rights say it will have a disastrous impact on women.

The liberal justices, led by Justice Ruth Bader Ginsburg, strenuously questioned the law, suggesting it was unnecessary and the procedure is safe.

Back in 2016, when Justice Anthony Kennedy was still on the bench, the court struck down a similar law out of Texas in a case referred to as Whole Woman's Health. Much has changed since then, however, as Kennedy has been replaced by Kavanaugh, who is considered more conservative on the issue.

Roberts, more than once, seemed to be trying to determine how bound the Court should be by that recent decision. Although he is a conservative justice, he has institutional concerns about the Court and may not like results to swing wildly just because the Court's membership has changed.

The Chief Justice questioned whether "results could be different" state by state but he also noted that an inquiry regarding the benefits of the law may not vary.

"I understand the idea that the impact (of the law) might be different in different places, but as far as the benefits of the law, that's going to be the same in each state, isn't it?" he asked.

Kavanaugh pushed a lawyer for the clinics. "Are you saying," he said, that such admitting privilege laws are "always unconstitutional"? He asked whether such a law could ever have a "valid" purpose.

Ginsburg pressed repeatedly on the notion that the law was not medically necessary. She noted that "most" abortions don't have complications. And she asked "what sense" it made to require a doctor to have admitting privileges in a hospital within 30 miles if a woman would more likely travel to a hospital near her own home if she needed to see a doctor after the abortion. Ginsburg noted that abortion is "far safer" than other procedures, including child birth.

Justice Elena Kagan said that doctors can be denied privileges under the law for reasons having nothing to do with their medical expertise. She, too, pushed on the claim that in one clinic, thousands of abortions had been performed, but the clinic had only ever transferred four patients to a hospital.

"Is it right that there is evidence in the record that Hope Clinic has served over 3,000 women annually for 23 years, so that's around 70,000 women, and has transferred only four patients ever to a hospital?" she asked.

The case will be decided by July, just as the presidential election is heating up.

If Louisiana prevails, it will likely embolden other states to pass similar laws.

"Disappointingly, we find ourselves back at the Supreme Court this term challenging a Louisiana law designed to shut down clinics," said Nancy Northup, the President of the Center for Reproductive Rights, the group behind the challenge. She charged Louisiana with "openly defying" the Supreme Court's 2016 ruling.

Louisiana's Unsafe Abortion Protection Act, blocked since its enactment in 2014 is an effort, state officials argued, to "improve abortion safety by means of doctor credentialing."

Louisiana Solicitor General Elizabeth B. Murrill says that clinics in the state have a "long disturbing" history of serious health and safety problems, abortion carries "known risks for serious complications" and that the act would bring abortion practice "into conformity" with the privilege requirements for doctors performing other outpatient surgeries. The penalty for violating the law is not more than $4,000 per violation.

As a threshold matter, Murrill argues that the justices shouldn't even get to the merits of the case because the doctors and the clinics bringing the case don't have the legal right -- or "standing" -- to be in court. Murrill notes that Louisiana women can challenge abortion regulations if they wish to do so -- "as individual women have done in numerous other abortion cases across the country" -- but that the clinics and doctors can't stand in their place. She said that's because the interests of a for-profit business that exchanges medical services for a fee might not align with patients seeking an abortion.

More than 200 members of Congress, including two Democratic House members, filed a brief in support of Louisiana on that point. They say there is an inherent conflict of interest between abortion providers and their patients when it comes to health and safety regulations.

"It is impossible for abortion clinics and doctors to share or represent the interests of their patients when they seek to eliminate the very regulations designed to protect their patients' health and safety," the lawmakers stated.

The Trump administration supports that position and argues that the law is valid . "Act 620 would not create a substantial obstacle to obtaining an abortion for a large fraction of Louisiana women seeking one -- let alone all such women," Solicitor General Noel Francisco told the Court.

The administration argues that the Court might need to narrow or overrule the Texas case from 2016 because it conflicts with the Court's older precedent.

The claims are rejected by lawyers for the Center for Reproductive Rights, which is representing two doctors and an abortion clinic in the state who say that if the law had been able to go into effect when it passed, it would have forced the closure of two of the state's three remaining clinics and left only one doctor with the ability to provide abortion care.

They argue the law serves no valid credentialing function and will force women to increase travel times and wait longer for the procedure. "Because Act 620 provides no benefits, it unduly burdens a woman's Fourteenth Amendment right to pre-viability abortion," Julie Rikelman, an attorney with the Center for Reproductive Rights, argued. She said that it is extremely hard to obtain admitting privileges in the state and that denials often have nothing to do with a doctor's competency to provide abortion care.

Rikelman also rejects the notion that only women seeking abortions could bring the suit, noting that a woman would only have a narrow time frame to bring such a suit and that such litigation often lasts for years. She stresses the close relationship between doctor and patient.

The appeals court took the unusual move of engaging in some fact finding, a task often handled by district courts, and concluded that the doctors had not made a good faith effort to get the credentials.

"There is insufficient evidence to conclude that, had the doctors put forth a good faith effort to comply with Act 620, they would have been unable to obtain privileges," the court held. "Instead of demonstrating an undue burden on a large fraction of women," the court said, the law "at most shows an insubstantial burden on a small fraction of women."

CORRECTION: This story has been updated to correct the spelling of Nancy Northup's name.

This story has been updated with details of oral arguments.