WASHINGTON—The Supreme Court, unable to resolve the dispute between religious employers and the Obama administration over contraception coverage in the government’s health-care law, sent the matter back to lower courts to seek a compromise between the parties.

The move Monday prolongs the four-year fight over whether the groups must offer contraception coverage under the Affordable Care Act. It also highlights how Justice Antonin Scalia’s February death has hobbled the ability of the court’s eight remaining members to resolve...