DENVER — Less than two weeks after a federal judge cleared the way for gay couples to wed in Utah, leaders from the socially conservative state on Tuesday asked the United States Supreme Court to halt the tide of same-sex marriages.

The move comes as Utah appeals a Dec. 20 ruling that overturned its voter-approved ban on same-sex marriage. While gay couples across Utah said the decision gave them a foothold as equal citizens of their state, state officials said the decision, by Judge Robert J. Shelby of Federal District Court in Salt Lake City, had created chaos while undercutting Utah’s right to limit marriage to one man and one woman.

In their filing on Tuesday, Utah officials said the state would suffer serious harm as same-sex marriages continued in county courthouses and government buildings across the state. Each one is “an affront to the sovereignty of the state and its people,” and “openly flouts” Utah’s ability to define what constitutes a marriage inside its borders, the state wrote.

Utah’s request landed on the desk of Justice Sonia Sotomayor, who handles cases in Utah and other Rocky Mountain States. She could rule on the state’s request for a stay, but is expected to refer the matter to the entire court. The justices could rule within days.