The
Morrill Anti-Bigamy Act specifically targeting Mormons was passed in 1862 that banned bigamy in federal territories. It was challenged by the Mormons, eventually resulting in the Supreme Court case decided in
Reynolds v. United States in 1879.
No, that's not quite right. The law and the resulting legal cases were tightly focused on banning the polygamy among Mormons. Mormons were not to commit bigamy, that is, they could not be married to more than one person at a time. The focus of the law and the Supreme Court decision was polygamy and whether or not the Mormons' religious liberty rights could supersede laws against bigamy. These cases were not concerned with the genders of the persons being married.
Most of the laws that built up around marriage had to do with inheritance and monogamous relationships, that is, one had to be divorced before marrying another. So yes, in a sense, they defined marriage, but they did not specifically ban citizens from getting married.