Two gay couples denied licenses have requested Rowan County Clerk Kim Davis be held in contempt of court.
But they asked a district judge to punish Ms Davis by fining her, rather than sending her to jail.
The US Supreme Court on Monday rejected the argument that her faith prevented her from carrying out her duty.
Despite court orders to continue issuing licences, Ms Davis turned away two gay couples seeking marriage licences on Tuesday morning.
Ms Davis, who cannot be removed from post because she is an elected official, stayed inside her office with the blinds closed as the couples demanded to see her.
She eventually came out, but she said she would not issue any licences.
"Under whose authority?" she was asked. "Under God's authority," she said.
Dozens of reporters and activists, both supporters and opponents of gay marriage, filled the waiting area of the Rowan County clerk's office, alternatively chanting "do your job" and "stand firm".
One of the men seeking a licence, David Ermold, said he and his partner would not leave the office until they were married.
"Then you're going to have a long day," Ms Davis replied.
Since the court legalised gay marriage nationwide two months ago, Ms Davis has refused to issue any marriage licences to anyone.
The American Civil Liberties Union sued her on behalf of two gay couples and two heterosexual couples.
The clerk has been ordered to appear in federal court on Thursday to address a motion filed by one of the couples asking for her to be held in contempt of court.
In making its ruling on Monday, the Supreme Court denied her request for a stay while she pursues an appeal.

Mr Jepson, who died in August, began his affiliation with the club in the 1960s and was named president in 2004.
The service took place at Headingley Stadium earlier.
Rhinos chief executive Gary Hetherington said: "Harry had a wonderful life and touched the lives of so many others."
Mr Jepson's love of the sport dated back to the 1920s when he first watched Hunslet at their Parkside Stadium.
Leeds players past and present were amongst those to pay tribute to Mr Jepson during the service.
Among those who spoke during the ceremony were Jamie Jones-Buchanan and Tony Currie, while former Leeds Rhinos and England captain Kevin Sinfield read Rudyard Kipling's poem 'If'.
The service was brought to a close by Lizzie Jones singing the Challenge Cup anthem 'Abide With Me' as the hearse carrying Mr Jepson's coffin left the ground for a family service.
Mr Hetherington said: "He knew people who had been involved in the formation of the game in 1895, and his death coincides with the 121st anniversary of the game he loved so much.
"He was a Rugby League legend, despite never playing the game and his passing is so sad."
A book of remembrance has been opened for fans in the club shop.
A Christian official has refused to issue marriage licences to same-sex couples in Kentucky despite exhausting all of her legal options.
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A memorial service has taken place to remember the life of Leeds Rhinos president and life-long rugby league fan Harry Jepson OBE, who died aged 96.