When advocates of same-sex “marriage” make statements about the importance of not denying a certain group access to the institution of marriage, they are being realists since they are implying (perhaps without realizing it) that marriage has a definite meaning that transcends the contingencies of culture and language. For if marriage did not have a fixed meaning that could in principle include certain classes of people, then there would be no way we could meaningfully speak of those classes being unjustly excluded, just as you could not meaningfully speak of tennis players being unjustly excluded from an athletic club unless you first knew that the category “athletic club” had a specific objective meaning that made possible the inclusion of tennis players in a way that it did not make possible the inclusion of, say, chess players or musicians.
In this respect, champions of same-sex “marriage” are realists. They think marriage has objective content that is fixed and specific, namely the union of two persons in some kind of love relationship. Because of this (so the argument goes) we should not exclude any person from such unions, whether that exclusion is based on race, gender, sexual orientation, or something else. The realism of this position coheres with the traditional view of marriage as a sexually complimentary union. The only difference is on the question of what marriage actually means: is it a union of sexually dimorphous persons or is it simply a union of persons?
In my experience, often the same champions of “gay marriage” who are realists when asserting that marriage is a union of persons will quickly switch to being nominalists when interacting with their opponents. That is, they will move from asserting that marriage has an objective meaning that is specific and fixed (namely the union of persons) to suggesting that the meaning of marriage depends entirely on arbitrary naming acts.
For example, in arguing against the notion that marriage involves an essential sexual dimorphism, defenders of gay marriage will frequently point to the fact that past cultures have sanctioned things like marriage to children or polygamy, as if this shows that marriage is entirely culturally relative, and entirely a matter of definition. In other words, marriage has no intrinsic meaning, but achieves its meaning through external will; therefore, the primary questions we should be looking at are questions about what social outcomes will result from defining marriage in a certain way.
The only problem is that if the champions of same-sex “marriage” were to be completely consistent with this nominalist turn, they would have to deny any objectivity to the claim that marriage is a union of persons in general (over and against a union specifically of a man and a woman), which mostly they are unwilling to do. This unwillingness is not surprising, because the entire argument about needing to “expand the pool of people eligible to marry” assumes that marriage has objective meaning and refers to a union of consenting adults who commit to romantic partnership and domestic life. But if marriage is an infinitely malleable social construct, then we cannot claim that its existence as the union of two persons is fixed and objective any more than we can claim that its existence as the union of a man and a woman is fixed and objective.