The lawsuit, filed in federal court in Ashland with the legal support of the ACLU, claims Rowan County Clerk Kim Davis has purposefully violated the plaintiffs' rights. The lawsuit also named the county.
Prior to the Supreme Court decision, same-sex marriages were banned, but Governor Steve Beshear directed county clerks to begin accepting those applications hours after the ruling.
The plaintiffs seek to create a class action suit to force Davis to issue licences to all eligible couples.
"When our laws are updated or changed, government officials have a duty and a responsibility to impartially administer those laws," Michael Aldridge, the ACLU state executive director, said in a statement.
Davis could not be reached to comment on Thursday, but in an interview with Reuters on Tuesday she said her religious beliefs prevented her from issuing marriage licenses to gay and lesbian couples.
In the lawsuit, ACLU legal director William Sharp wrote that Davis' religious conviction "is not a compelling, important or legitimate government interest."
Davis said she knew a lawsuit was likely coming.
"I've thought a lot about what's going to happen and what to do," she said, when asked if she's considered resigning. "I've just been praying about it. You know, if this is a fight I need to stay and fight, then I'll fight. If it's something that's bigger than me and bigger than everyone else, we'll just see about it when it comes."
One of the attorneys who filed the lawsuit, Laura Landenwich, wrote that Davis "has the absolute right to believe whatever she wants about God, faith, and religion, but as a government official who swore an oath to uphold the law, she cannot pick and choose who she is going to serve, or which duties her office will perform based on her religious beliefs."
While the couples who filed suit could have applied for a licence elsewhere in the state, Aaron Skaggs, a plaintiff in the case, said it was important to be accepted in his own community.
Rowan is not the only Kentucky county that has stopped issuing marriage licences. Billy Joe Lowe, the clerk of Green County, told WFPL-FM on Thursday that when his office resumes issuing licences next week, it will not accept applications from same-sex couples.