In fact same-sex marriages do not differ from ordinary marriages in the USA. Same-sex marriages were legalized throughout the United States on June 26, 2015, when the Supreme Court held a decision on the case Obergefell v. Hodges. The United States Supreme Judicial Authority held that the United States Constitution guarantees the right to same-sex couples to marry. The Supreme Court of the USA bound the authorities of all the states to register same-sex marriages and recognize their validity performed in other jurisdictions.
Typically, same-sex couples take such a way: a US citizen files Form I-129F (fiancé visa). As in former USSR countries same-sex marriages are illegal, it is easier to enter the USA with fiancé visa and register the marriage on the territory of the United States. If the marriage has already been registered (for example, some people do it in Amsterdam), a US citizen or a green card holder can file Form I-130 (petition for alien relative).