Let's include in any list of reproductive options for gay men or lesbians seeking to reproduce good old copulation. Just because it is not enjoyable does not necessarily mean it is not possible. Edward III got made somehow, okay? and not by Mel Gibson.
Also, let's not forget that continued immigration could keep the population of such a state constant even with a relatively low birthrate. So long as there's a Utah...
Seriously though, obviously the gay nationalism concept held much greater attraction for gay men and lesbians in the United States and Europe back in the days when social and legal prohibitions were much stronger and homosexuality did actually mean enduring discrimination, harassment and deprivation of freedom in such a way as to seriously inhibit one's daily life. By which I mean suffering through electroshock therapy, enduring a prison term, or being denied a law license for being a homosexual. As matters stand today, I doubt I would migrate to a gay and lesbian specific state if one existed.
Of course matters are far different in places like Russia, Africa or the Arab world. Now someone mentioned the possibility of gay men, lesbians and transgender people in these places immigrating to countries more tolerant of them. Of course this can be easier said than done, though many countries grant asylum for persecution on the basis of sexual orientation.
However, if someone were to try to imagine how the gay nationalism concept might work in the twenty-first century, they might think of creating a place to which sexual dissidents in oppressive societies can emigrate quickly and easily, without paying substantial fees to an immigration lawyer or lingering on a waiting list.
You'd need a place not highly populated, relatively poor, relatively easily defensible, and not claimed by some other place with a strong military.