Whatever the choice, no plan can be turned around in two weeks, or two months. It took more than two years for the administration to figure out how to provide contraceptive coverage for women at nonprofit groups that have religious objections. That arrangement allowed religious organizations to fill out a form that would transfer the delivery of free coverage under the Affordable Care Act to others.

But many of the nonprofit groups say that even notifying an insurer of their objections through the opt-out form would make them complicit in a moral wrong. Some consider all contraception to be wrong; others object only to devices and drugs like the so-called morning-after pill that they believe may cause abortions. One such objector was Wheaton College, a Christian liberal arts school in Illinois, and the Supreme Court granted it a temporary exemption in the ruling on Thursday.

That move divided the court along gender lines, with Justices Ruth Bader Ginsburg and Elena Kagan joining Justice Sotomayor’s unsparing dissent. They said the court majority had endorsed the opt-out form just three days earlier in the Hobby Lobby case, in which Justice Stephen G. Breyer joined the three female justices in dissent in the 5-to-4 ruling.

The court’s conservative majority — all men — was sanguine about the availability of other ways for the administration to deliver coverage for every form of birth control approved by the Food and Drug Administration.

Image A Hobby Lobby store in Laurel, Md. The company’s ties to religious principles resulted in a case before the Supreme Court. Credit... Drew Angerer for The New York Times

Yet officials are struggling to make sense of a sunny sentence in the court’s order on Thursday exempting Wheaton from the opt-out form. “Nothing in this interim order affects the ability of the applicant’s employees and students to obtain, without cost, the full range of F.D.A.-approved contraceptives,” the majority said in the unsigned opinion.

It said Wheaton could merely notify the government of its religious objections in writing rather than send the opt-out form to its coverage providers.