I’m a U.S. citizen male, living in Texas, in a long-term relationship with another man, who lives in Costa Rica. We’ve done a lot of traveling back and forth over the years, but never been able to settle down in the same country. And Texas didn't allow same-sex marriage. But recent Supreme Court decisions have been very exciting – we’d like to get married and have him join me in the United States. Would a fiance visa work for that?
A fiance visa (K-1) is indeed available to same-sex couples, based on current immigration law (and aided by Supreme Court decisions in 2013 and 2015).
As with other fiances, you would need to prove that you are a U.S. citizen, that the two of you have met within the last two years, that you plan to marry within 90 days of your intended’s entry into the United States, and that your partner is not inadmissible to the U.S. for health, crime, security, past unlawful presence in the U.S., or other reasons.
For more on the requirements and procedures, see the “K-1 Fiance Visas” section of Nolo’s website.
After your fiance arrives in the U.S., you will need to get married reasonably soon, in order to prepare to submit your new spouse’s adjustment of status (green card) application. Based on the Supreme Court's 2015 Obergefell decision, same-sex marriage should be possible in every U.S. state -- but if you're reading this in mid-2015, you might want to double-check that the state where you intend to marry is fully geared up for this, or prepare for some long lines at City Hall!