The federal government and an order of Catholic nuns will face off against several states June 6 over Obamacare’s birth control coverage requirement.

Each side will get 30 minutes to persuade the U.S. Court of Appeals for the Ninth Circuit of the validity—or invalidity—of final agency rules allowing employers who object on religious or moral grounds to opt out of providing contraceptive coverage in their employee health plans.

About 62.8 million women are eligible for contraceptive coverage under the Affordable...