Rowan County Clerk Kim Davis stopped issuing all marriage licences the day after the US Supreme Court legalised same-sex marriage.
The American Civil Liberties Union has sued her on behalf of four couples, two gay and two straight.
In court on Monday in Covington, Kentucky, Ms Davis said issuing the licences would violate her religion.
"If... I authorise that I'm saying I agree with it, and I can't," she said.
The state's governor, Steve Beshear, said county clerks must issue marriage licences to same-sex couples immediately after the Supreme Court's ruling.
Two clerks in the state refused. They can only be removed from office by the state legislature.
US District Judge David Bunning will not rule on Ms Davis' case until August, he said.
Mr Bunning could order Ms Davis to issue licenses, impose a fine or put her in jail.
Ms Davis wants Mr Beshear to pass a law protecting county clerks and allow people to get marriage licences online.
The governor refused to call the legislature back into session to do so.
A Christian official has defended her refusal to issue marriage licences to same-sex couples in Kentucky.