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.

Rotherwas Chapel, in Herefordshire, has been unused for many years and currently can only be viewed by making an appointment through English Heritage.
But a newly formed Friends of the Chapel group are striving to make the historic chapel, once part of a large estate, more accessible to visitors.
The mass, open to all, is due to take place on Monday.
Margot Miller, a member of the Friends of the Chapel, which formed earlier this year, said research had not thrown up any records of mass ever being held there.
"It was originally used as a family church for the Bodenham family, not a parish church, so we don't know if any mass was heard," she said.
The Ministry of Defence took over the building after the death of the last Bodenham family member and it was used by soldiers defending the munitions factory in Rotherwas.
The rest of the estate was knocked down.
English Heritage eventually took the chapel over and Ms Miller and other volunteers formed a group to help look after and clean the building.
"It is a lovely chapel," she said.
"A beautiful little building. We think it was mainly used in the 16th century and it has a little altar and stained glass windows. The Bodenham family spent a lot of money on it."
The first mass is designed to coincide with celebrations for the Feast of the Assumption. The chapel is called Our Lady of the Assumption.
Plans for a memorial for Henry Bodehnam, killed in France in 1916, are being arranged for September and it is also hoped a carol service will be held in December.

The company has introduced an option to allow you to flag a story as "purposefully fake or deceitful news".
This should affect the distribution of that story, making it less likely to appear in your feed - but it won't be taken off Facebook altogether.
Hoaxes have tricked users in the past and with 1.35bn currently on Facebook, that's a lot of people to fool.
"A post with a link to an article that many people have reported as a hoax or chose to delete will get reduced distribution in the News Feed," Facebook explained.
Examples of recent fake stories include dinosaur sightings and research claiming to prove the existence of Santa Claus.
Earlier this month users again started posting privacy notices - that's a response to a hoax from two years ago.
In the past, Facebook has been criticised for not doing enough to tackle illegitimate accounts; in 2012 the company admitted that more than 80m were likely to be fake.
At the time, the company said that "undesirable" accounts included those using fake names which were "intended to be used for purposes that violate our terms of service, such as spamming".
About a third of adults in the US now use Facebook to get news updates, according to a 2013 study by the Pew Research Center in collaboration with the John S and James L Knight Foundation.
Fans of sites like The Onion should be reassured by Facebook's claim that "satirical" articles, "intended to be humorous, or content that is clearly labelled as satire," will not be affected.
Follow @BBCNewsbeat on Twitter, BBCNewsbeat on Instagram and Radio1Newsbeat on YouTube
A Christian official has defended her refusal to issue marriage licences to same-sex couples in Kentucky.
[NEXT_CONCEPT]
A little-used chapel is to host what is thought to be its first public mass.
[NEXT_CONCEPT]
Hoaxes and fake stories in your News Feed should be a thing of the past, if Facebook's updated algorithm works.