“Kate desperately wanted to be able to get care where she lives and recover at home surrounded by family,” Nancy Northup, the chief executive for the Center for Reproductive Rights, which was representing Ms. Cox in her case, said in a statement.

The case was believed to be the first to seek a court-ordered exception since the U.S. Supreme Court overturned Roe v. Wade last year, clearing the way for Republican-controlled states like Texas to enact near-total bans on abortions.

It marked a new chapter in the legal history of abortion in the United States, with pregnant women now going to court seeking permission for their doctors to do what they determine to be medically necessary without fear of severe criminal or civil penalties.

That case, Zurawski v. Texas, involves women who said they were forced to continue pregnancies, despite dangers to their health, because the vagueness of the state’s exemptions made doctors extremely cautious about when a medical condition was serious enough to allow for an abortion.

The judge issued a temporary restraining order barring Mr. Paxton and others from enforcing the state bans against Dr. Karsan, Ms. Cox’s husband, and any medical staff members who assisted an abortion in her case.

Mr. Paxton objected, first sending letters to three Houston hospitals where Dr. Karsan can admit patients, saying that the judge’s order was only temporary and would not protect them from civil or criminal penalties if they allowed Ms. Cox’s procedure.

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