The Mississippi case has been regarded as a bellwether for abortion rights, challenging precedents that set viability—the fetus’s ability to survive outside the womb—as the baseline for expanded government power to restrict the procedure. In its initial appeal, filed in June 2020, Mississippi argued its Gestational Age Act complied with existing precedent, suggesting that Roe be overruled only if the court found no other way to uphold the state law.

But the brief filed Thursday reframed the arguments, taking direct aim at the 1973 Roe decision and a 1992 follow-up, Planned Parenthood v. Casey, that narrowed Roe’s scope. Under current precedent, regulations that place an undue burden on women seeking abortion are unconstitutional.

“Roe and Casey are egregiously wrong. The conclusion that abortion is a constitutional right has no basis in text, structure, history, or tradition,” Mississippi argued. “Roe broke from prior cases, Casey failed to rehabilitate it, and both recognize a right that has no basis in the Constitution.”

Even if there were policy reasons to allow women the right to end pregnancy in 1973, the state argued that times had changed.

“The march of progress has left Roe and Casey behind,” Mississippi said in the brief. “Those cases maintained that an unwanted pregnancy could doom women to ‘a distressful life and future,’ that abortion is a needed complement to contraception, and that viability marked a sensible point for when state interests in unborn life become compelling.”

But “today, adoption is accessible and on a wide scale women attain both professional success and a rich family life, contraceptives are more available and effective, and scientific advances show that an unborn child has taken on the human form and features months before viability,” the state argues.

“If Roe falls, half the states in the country are poised to ban abortion entirely,” said Nancy Northup, president of the Center for Reproductive Rights, which represents the Jackson Women’s Health Organization in the case. “Women of childbearing age in the U.S. have never known a world in which they don’t have this basic right, and we will keep fighting to make sure they never will,” she said.

When Mississippi filed its initial appeal in June 2020, the court’s direction hung on the vote of Chief Justice John Roberts, who has been skeptical of abortion rights but, in recent cases, reluctant to discard wholesale the precedents recognizing them. With the court otherwise divided between other conservatives hostile to abortion rights and four liberals committed to their defense, antiabortion activists typically framed litigation to chip away at precedents rather than overruling them.

The September death of Justice Ruth Bader Ginsburg changed the calculus, allowing then President Donald Trumpto appoint Justice Amy Coney Barrett, who in private life had belonged to an antiabortion organization. As is typical for judicial nominees, Judge Barrett said during her confirmation hearing that her personal views wouldn’t influence her approach to deciding cases.

During a 2016 campaign debate, Mr. Trump said were he given three Supreme Court appointments, Roe would be overruled “automatically.” With Justice Barrett joining Justices Neil Gorsuch and Brett Kavanaugh, the Mississippi case now is poised to test Mr. Trump’s prediction. Two other justices, Clarence Thomas and Samuel Alito, have consistently voted against abortion rights.

In March Mississippi Attorney General Lynn Fitch, a Republican, hired Scott Stewart, a veteran of the Trump Justice Department and former law clerk to Justice Thomas, as state solicitor general. Mr. Stewart is expected to argue the case.

The opposing brief from the Jackson Women’s Health Organization, the state’s only abortion clinic, is due Sept. 13. In an initial brief filed last August, the clinic argued that 50 years of precedent had established viability as a clear marker where the balance between a woman’s rights over her body begin to yield to a state’s interest in protecting potential life.

“Before viability, the State’s interests, whatever they may be, cannot override a pregnant person’s interests in their liberty and autonomy over their own body,” the clinic argued.

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