I don't believe the government should recognize interpersonal relationships, either. Any interpersonal relationships.
Families form economic units, and marriage is a contractual commitment. Since property rights and inheritance rights are affected by such commitments, the government has a legitimate role to play in regulating them. One of the reasons to support homosexual liaisons is to guarantee inheritance rights, visitation rights, estate management rights, and so forth to spouses. I cannot imagine why the government should refuse to recognize such liaisons. As a committed libertarian, you probably don't agree that the government should regulate business and contractual relationships to the extent that it does, and that is an area where we probably have strong disagreements.
...Homosexuality violates my religion's tenets, but it's likely that the homosexual doesn't hold with my religion.
I might counsel against engaging in homosexuality, but would demonstrate against a law banning homosexuality.
It is almost certain that some members of your religion are homosexual, and some will probably work to get religious leaders to incorporate tolerance in religious doctrinal policies. In my view, all religious beliefs are social conventions that reflect the attitudes of members. Our gods do what we tell them to do, and they seldom stray far from our prejudices. So it is natural for most religions to seek to suppress homosexuality, since that expresses a popular lack of tolerance.
Such prejudice is itself immoral, IMHO, because it does greater social harm than social good. And many religious leaders preach tolerance of homosexuality. However, most people seem to reject appeals for social tolerance in this area. Rejection of homosexuality is a powerful "family value" advantage to political conservatives in US elections, and it will be exploited to the detriment of people of my political persuasion.