Just weeks after the U.S. Supreme Court struck down part of a federal law that defined marriage as a union between a man and a woman, judges in lower courts are citing and even building on the ruling in battles over state laws concerning same-sex marriage and other issues affecting gays and lesbians.

A federal judge in Cincinnati cited the June ruling in U.S. v. Windsor in finding Ohio's 2004 law banning the recognition of same-sex marriages from other states "likely" to be unconstitutional, at least in regard to...