Sotomayor responded by setting a deadline of by noon Friday for legal briefs from same-sex couples. She can act by herself or get the rest of the court involved.
"Numerous same-sex marriages are now occurring every day in Utah," Utah lawyers complain in the filing. "Each one is an affront not only to the interests of the state and its citizens in being able to define marriage through ordinary democratic channels, but also to this court's unique role as final arbiter."
Also Tuesday, the Utah Attorney General's Office advertised a formal bid request to outside law firms for help preparing the appeals.
State officials have said it could cost $2 million, bringing criticism from a lawyer for couples who sued to overturn the ban and say Utah should give up the fight.
"We are disappointed that Utah will spend millions of dollars in taxpayer's money, to attempt to reinstate laws which deny due process and equal protection to all of Utah's citizens," the lawyer, James Magleby, said Tuesday.
Utah insists that states have the authority to define marriage as between a man and woman. "That states have a powerful interest in controlling the definition of marriage within their borders is indisputable," Utah said in the filing.
U.S. District Judge Robert Shelby's decision on Dec. 20 came as a shock to many in the state, which approved the ban on same-sex marriage in 2004.
Shelby and the 10th U.S. Circuit Court of Appeals have already refused to halt weddings while the state appeals what it calls a "rush to marry." Magleby argued that because so many couples have already married, "there is no emergency need for a stay."
Shelby's decision came late on a Friday afternoon and sent people rushing to a county clerk's office in Salt Lake City — about 3 miles from the headquarters of the Mormon church — for marriage licenses. The following Monday, 353 more gay and lesbian couples grabbed a license, some camping out overnight to get in line early the next morning.
After the 10th Circuit Court refused to halt the ruling, the few county clerks who had refused to issue licenses changed course. Officials say things have slowed down after a run on marriage licenses that started hours after Shelby's decision.
Since then, Gov. Gary Herbert has directed state agencies to comply with Shelby's order, meaning gay couples are eligible for food stamps and welfare, among other benefits. The state Tax Commission said it was looking at changing tax returns to allow same-sex couples to file jointly, although it didn't immediately give assurances that will happen.
Nearly two-thirds of Utah's 2.8 million residents are members of The Church of Jesus Christ of Latter-day Saints, and Mormons dominate the state's legal and political circles. The Mormon church was one of the leading forces behind California's short-lived ban on same-sex marriage, Proposition 8.
The church says it stands by its support for "traditional marriage" and hopes a higher court validates its belief that marriage is between a man and woman.
In the papers filed Tuesday, Utah argues that children are best raised by a mother and father in a good relationship.
"On average children navigate developmental stages more easily, perform better academically, have fewer emotional disorders and become better functioning adults when reared in that environment," it says.