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Kim Davis, right, the clerk of Rowan County, Ky., speaks with David Moore after refusing to issue him a marriage license on Tuesday.Credit Timothy D. Easley/Associated Press

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LEXINGTON, Ky., — A county clerk in Kentucky who objects to same-sex marriage on religious grounds denied licenses to gay couples on Tuesday, just hours after the Supreme Court refused to support her position.

In a raucous scene in the little town of Morehead, two-same-sex couples walked into the Rowan County Courthouse, trailed by television cameras and chanting protesters on both sides of the issue, only to be told by the county clerk, Kim Davis, that she was denying them marriage licenses “under the authority of God.”

A deputy clerk told two couples, April Miller and Karen Roberts, and David Ermold and David Moore, that no licenses would be issued Tuesday. Ms. Davis at first remained in her office with the blinds drawn, but emerged after the applicants demanded to speak with her.

“Tell her to come out and face the people she’s discriminating against,” Mr. Ermold shouted.

When Ms. Davis asked the applicants to leave, Mr. Moore said he would remain in the clerk’s office until he and Mr. Ermold had a license. “Then you’re going to have a long day,” she replied.

Ms. Davis stopped issuing all marriage licenses after the Supreme Court legalized same-sex marriage in June.

After the state’s governor told county clerks to issue marriage licenses to all eligible couples, Ms. Davis filed suit in federal court, arguing that she should be excused from the obligation, given her religious beliefs. A District Court judge ruled against her, as did the Sixth Circuit Court of Appeals, and she appealed to the Supreme Court.

On Monday, a stay granted by the District Court expired, and the Supreme Court rejected without comment Ms. Davis’s emergency application for a new stay, pending the outcome of her appeal. That left her no legal grounds to refuse to grant licenses to same-sex couples.

“She’s certainly in contempt of court by any definition of the term, so the District Court has an array of sanctions it can resort to, to deal with that,” said Daniel J. Canon, a lawyer for some of the same-sex couples seeking licenses. “It can levy civil or criminal sanctions against her, and we had hoped that it would not come down to that.”

Ms. Davis told the Supreme Court that her Apostolic Christian faith forbade her to affix her name to a document endorsing the view that the marriages of gay men and lesbians were authentic. “This searing act of validation would forever echo in her conscience,” her lawyers told the court.

A fellow Kentucky clerk, Casey Davis, who has protested alongside Ms. Davis, insisted Tuesday that “we’ve not tried to prevent,” same-sex marriages, “we’ve only tried to exercise our First Amendment rights.”

Mr. Davis said same-sex couples could go to another county to get married when a county clerk objects due to religious beliefs. “There was a lot of people that died for that right and I think we should be able to exercise it,” Mr. Davis told CNN. He is riding his bicycle from Louisville, Ky., to the West Coast to raise awareness for what he described as an assault on religious liberty.