WASHINGTON—The Trump administration said Friday it is paring back an Obama-era requirement that employer-provided health benefits cover contraception, re-energizing a volatile debate on the role of government and religion in health care.

The new rules, announced by the Health and Human Services Department, immediately grant a sweeping exemption to employers who object to the mandate that employers cover contraception without copayments or other out-of-pocket costs. Employers could now omit coverage of some or all contraceptives...