I have three main objections to homosexual "marriage", so called.
The first is that I believe homosexual activity is (theologically) a sin and (biologically) quite different from heterosexual intercourse - you know, in the part where it can't result in children. The church institutional, I am confident, will continue to recognize (and even publicize) the distinction, but any time there are significant differences between the proscriptions of one part of society, and society officially as a whole, trouble tends to follow.
So stemming from that, the second is that I worry that, if the government recognizes homosexual couples as "married", the forces of Caesar will quickly be called on to try to force the church into line. This possibility was always present, but has become infinitely harder for activists to disavow with the current administration's dedication to running roughshod over religious (and other) freedoms in the name of "women's reproductive health".
The last is that the language involved makes no sense. If marriage is the recognition of a formal (and spiritual) union between a man and a woman, then talking about "homosexual marriage" is ludicrous, because of the bit where there's not a woman (or a man). If marriage is merely the formalization of a sexual relationship between two people - which, to follow the modern fads, is not actually morally binding and can of course be dissolved at any point by divorce - then what is the point of marriage at all? If you appeal to "family" - which is to say, children - that is not a point in favor of the homosexuals, because homosexual sex does not result in children. True, such a couple could adopt or use some other method to, ah, acquire them. But that doesn't need a marriage, that I am aware of.
In short, I am against homosexual marriage because the only possible purpose of such a policy would be to attempt to legitimize something I do not think is legitimate, and - even more of a concern to my logical mind - equate two things I do not think are equal.