In Kentucky, Judge Heyburn ruled in one of many cases across the country brought against state bans on same-sex marriage in the wake of a Supreme Court ruling last year throwing out a key part of the federal Defense of Marriage Act.

Judges in Oklahoma, Utah and Virginia have overturned state bans on grounds that they violate the Constitution’s guarantee of equal protection to all under the law. The rulings have been stayed as the cases wend their way through appeals courts.

The ruling in the Kentucky case, which challenges a constitutional amendment defining marriage as between one man and one woman, struck down only part of the law. A second case before the same judge seeks its full overturning.

With Mr. Conway the best-known likely candidate to succeed Mr. Beshear, there was much speculation among Kentucky political analysts about how his stance might be received by the state’s socially conservative voters.

The ban on same-sex marriage was approved by 75 percent of voters in 2004.

But as acceptance of same-sex marriage has grown in Kentucky, as elsewhere, it may prove less a wedge issue. Hal Heiner, a Republican who announced his candidacy for governor on Tuesday, made no immediate criticism of Mr. Conway.

“When it comes to cultural trends, we are behind the curve, but the curve is clearly moving here,” said Al Cross, a former chief political writer for The Courier-Journal of Louisville. He said Mr. Beshear, who is legally prevented from seeking another term, was giving cover to his fellow Democrat.

“If you had the top two Democratic officials saying we are not going to appeal this ruling that so many people in our state find objectionable, that would be bad for the Democratic brand,” he said.