(CNN) -- The U.S. Supreme Court on Tuesday refused to stop Texas from implementing a part of a new abortion law that requires doctors to get admitting privileges at a hospital within 30 miles of the clinic where they're providing abortion services.
A federal appeals court had reinstated the key part of the law, which is considered among the most restrictive in the country.
Abortion-rights groups, which say more than a third of centers that had provided abortions in Texas have since stopped, then asked the Supreme Court to overturn the appeals court ruling.
"We may not vacate a stay entered by a court of appeals unless that court clearly and 'demonstrably' erred in its application of 'accepted standards,'" Justice Antonin Scalia wrote in support of Tuesday's order.
"Reasonable minds can perhaps disagree about whether the Court of Appeals should have granted a stay in this case. But there is no doubt that the applicants have not carried their heavy burden of showing that doing so was a clear violation of accepted legal standards -- which do not include a special 'status quo' standard for laws affecting abortion," he wrote.
Four justices -- Stephen Breyer, Ruth Bader Ginsburg, Sonia Sotomayor and Elena Kagan -- dissented.
The original lawsuit, filed in U.S. District Court in Austin by Planned Parenthood on behalf of more than a dozen women's health care providers across Texas, alleged the state's new abortion law violates the constitutional rights of women and puts unreasonable demands on doctors who perform abortions.
The bill originally failed to gain approval because of a Democratic filibuster led by state Sen. Wendy Davis.
Gov. Rick Perry then called the Legislature into a second special session to continue consideration of the bill.
Davis, who is running for governor, blasted Tuesday's decision.
"Clinics will close and women's health will be hurt because of this law. This is an abuse of power by politicians in Austin. I trust women to make their own decisions and will continue to work to make sure that women and mothers are safe and have access to adequate health care," she said in a statement.
Perry cheered the ruling.
"This is good news both for the unborn and for the women of Texas, who are now better protected from shoddy abortion providers operating in dangerous conditions. As always, Texas will continue doing everything we can to protect the culture of life in our state," he said.
CNN's Dave Alsup, Matt Sloane and Adam Levine contributed to this report.