Citizens of the United States, as well as lawful permanent residents, can sponsor their spouses for immigrant visas, or "green cards." Ever since the Supreme Court struck down a section of the Defense of Marriage Act in 2013, this sponsorship has been allowed for same-sex spouses. Of course, the legalization of same-sex marriage in every state in the 2015 Obergefell v. Hodges case has created more demand for same-sex spouses to petition for green cards.
Further, the U.S. Citizenship and Immigration Services agreed to re-review any petitions that were rejected prior to the DOMA decision. There is no charge for asking that the case be reopened or for USCIS to review previous petitions. In some cases, it may be preferable to file a new petition with additional supporting documentation; there are fees involved with that.
All green card applications are reviewed by the U.S. Citizenship and Immigration Services to make sure that the foreign person is eligible. However, people cannot legally be denied a green card because their spouse is of the same gender.
There are advantages for people in "green card marriages" when it comes to the time frame for becoming a naturalized U.S. citizen. Generally, people who are in this country as lawful permanent residents must wait about five years before they're eligible to be naturalized citizens. Those who are in a "marital union" with a U.S. citizen only have to wait three years for naturalization.
Even though federal law has superseded state laws when it comes to granting green cards to same-sex spouses, for people in states like Georgia, which didn't recognize same-sex marriage as legal until the 2015 SCOTUS decision, same-sex couples can still face challenges in dealing with immigration issues (and a raft of other legal issues.) An experienced Georgia immigration attorney can advise you when seeking a green card for your spouse.
Source: FindLaw, "Same Sex 'Green Card Marriage' Facts," accessed Oct. 21, 2016