DENVER — The push for same-sex marriage, which has celebrated victory after victory in courtrooms across the country, entered an uncertain stage on Thursday as a federal appeals court appeared divided about whether the socially conservative state of Utah could limit marriage to a man and a woman.

In an hour of arguments inside a packed courtroom, three judges from the Federal Court of Appeals for the 10th Circuit sparred with lawyers about how such bans affected the children of same-sex parents and whether preventing gay couples from marrying actually did anything to promote or strengthen heterosexual unions and families.

The three judges — two appointed by Republican presidents and one by a Democrat — focused on fine-grained issues, like what level of judicial scrutiny to apply to the case, as well as more profound questions of how to define a marriage and whether state bans on same-sex nuptials were akin to those against polygamy, or instead fundamentally violated the constitutional rights of same-sex couples.

While the decision will reverberate across Utah, it will hardly be the last word on whether same-sex couples have the same rights to marry as heterosexuals. Next week, the same appeals court is scheduled to hear arguments over Oklahoma’s ban on same-sex marriage, which a federal judge declared unconstitutional in January. Marriage cases in several other states, including Virginia and Texas, are percolating through the courts, and the Supreme Court is widely expected to tackle the issue.