I’ve just proposed to my same-sex boyfriend, who lives in Japan. He said yes! He already has a multiple entry U.S. tourist visa, so I’d love it if he could just get on a plane tomorrow and come here to hold the wedding and get a green card. I’ve read that people who enter the U.S. legally and marry U.S. citizens (which I am) can simply apply for adjustment of status. Is it that easy?
NO!!! What you’re describing is considered visa fraud, and it’s the one thing most likely to mess up your plans at this point.
It is true that same-sex marriage is a valid ground for U.S. immigration, and true that, once you are lawfully married in a place where same-sex marriage is legal, your boyfriend becomes your “immediate relative” and potentially eligible for a marriage-based U.S. green card. (Same-sex marriage became legal in the U.S. in 2015, as a result of the Supreme Court's decision in Obergefell v. Hodges.)
But notice we said “potentially.” Legal marriage or no, he also has to be “admissible” to the United States. Visa fraud is a ground of inadmissibility.
Using a visitor visa (which requires “nonimmigrant intent,” or the plan to leave the U.S. at the end of the permitted period of stay) in order to enter the U.S. with “immigrant intent” (the plan to marry and get a green card) qualifies within this ground of inadmissibility and your boyfriend's green card application could be denied as a result. (Unless he qualifies for a waiver, but don't count on it.)
For a full explanation, see Risks of Entering the U.S. as a Tourist, Then Applying for Marriage-Based Green Card.
A better option might be apply for a K-1 fiancé visa, or for the two of you to get married abroad in a country that allows same-sex marriage and then apply for a marriage-based immigrant visa to the United States. Consult an immigration attorney for a full analysis.
A final note: You might be wondering how you got the idea that this plan would work, or even know people who entered as tourists then got a green card based on marriage to a U.S. citizen. The key is the noncitizen's plans upon U.S. entry. Someone who enters the U.S. truly meaning to be a tourist, and then, say, falls in love with and marries a U.S. citizen; or even decides to marry a U.S. citizen whom he or she already knew; is in a different category and might succeed in getting a U.S. green card through adjustment of status. Even so, if the marriage takes place within the first three to six months of the person's visit, U.S. immigration authorities are likely to raise questions about immigrant intent.