If state and local officials dig in their heels and resist complying with the court’s ruling to issue marriage licenses to same-sex couples, the Justice Department may have to take legal action, he said.

“It took a period of time after other major civil-rights holdings by the Supreme Court before all states came into compliance. So yes, there’s potentially a role for the Justice Department to play,” Mr. Holder said.

“My hope would be that it doesn’t come to that,’’ he continued, adding that federal officials should wait a few months to let the ruling sink in before taking action.