Bryant, in a statement late Tuesday, said he was disappointed by the court's ruling.
"This measure is designed to protect the health and safety of women who undergo this potentially dangerous procedure, and physicians who provide abortions should be held to the same standards as physicians who perform other outpatient procedures," Bryant said.
Ten states have adopted similar laws, forcing a growing number of clinics to close. Many hospitals ignore or reject abortion doctors' applications, and won't grant privileges to out-of-state physicians. Both obstacles were encountered by the traveling doctors who staff Mississippi's last open clinic, the Jackson Women's Health Organization.
The ruling from the conservative 5th Circuit was narrowly crafted to address the situation in Mississippi, but it could have implications for the other states with similar laws and dwindling access to abortion, such as Wisconsin and Alabama, whose officials have said women could cross state lines if clinics close, said the center's litigation director, Julie Rikelman.
Attorneys for Mississippi argued that if the state's last clinic closed, women could still get abortions in other states. But the judges said the U.S. Supreme Court's 1973 Roe v. Wade decision established a constitutional right to abortion for all citizens—and that Mississippi may not shift its obligations to other states.
"Pre-viability, a woman has the constitutional right to end her pregnancy by abortion," wrote judges E. Grady Jolly of Mississippi and Stephen A. Higginson of Louisiana. The law signed by Bryant "effectively extinguishes that right within Mississippi's borders," they said.
Supporters of admitting-privileges laws say they protect women's health by ensuring that a physician who performs an abortion in a clinic would also be able to treat the patient in a hospital in case of complications.
Opponents say the requirement is unnecessary, since patients in distress are automatically treated in emergency rooms, and that it gives religious-affiliated hospitals veto power over who can work in an abortion clinic and, by extension, whether a clinic can stay open.
"Today's ruling ensures women who have decided to end a pregnancy will continue, for now, to have access to safe, legal care in their home state," Center for Reproductive Rights president Nancy Northup said Tuesday.
A different panel of the 5th Circuit, which handles cases from Mississippi, Louisiana and Texas, upheld a 2013 Texas law requiring physicians to have admitting privileges at a hospital within 30 miles. In that March ruling, the judges said traveling fewer than 150 miles to obtain an abortion is "not an undue burden."
Even now, women from Iuka, Mississippi, in the state's northeast corner, need to drive 280 miles to reach Jackson.
The clinic remains open, using out-of-state physicians who travel to Mississippi to do abortions several times a month. For years, the clinic has had an agreement with a local physician who will meet a patient at a Jackson hospital in case of complications. Clinic owner Diane Derzis has said such complications are rare.