In terms of actual prohibitions:
1- Male homosexual sex is prohibited to both Jew, and under the covenant G-d made with Noah, to non-Jews as well.
2- Male homosexual intimacy that is not prohibited under #1 is prohibited to Jews under the same kinds of laws that prohibit masturbation.
3- At least some forms of female homosexual sex if not all are prohibited by the Torah to Jews. (See Rashi, Yevamos 71a; Rambam Issurei Bi'ah 21:8)
4- There is a prohibition for both Jews and non-Jews to contract a marriage between two men, two women, and one that includes a woman and more than one man. This is derived from the verse "כְּמַעֲשֵׂה אֶרֶץ-מִצְרַיִם אֲשֶׁר יְשַׁבְתֶּם-בָּהּ, לֹא תַעֲשׂוּ -- as per the actions of the Land of Egypt which you live in, do not do" (see Rambam ibid.) Notice this prohibition condemns Egypt for contracting such marriages, because the prohibition is on non-Jews as well as Jews.
There is never a prohibition against a desire. (The topic of the 10th commandment and whether it is actually against the feeling of jealousy would take us too far afield, but it is NOT generally taken that way among commentaries.)
So, there is no prohibition against desiring gay relations. Nor against wanting to live monogamously with a partner of the same sex. Acting on the desire is the sin; not having it.
BUT, there is still a difference between having a desire, and self-identifying as someone who has it. In other words, the only kind of "Gay Pride" that would be consistent with the Torah would be pride in the knowledge that Hashem thought one was capable of the challenge. Most of us are not, and therefore spared the whole battle against a desire for this particular prohibition. The fact that Hashem thought someone's soul could handle that kind of work says something. The bottom line is, though, that it's a desire that requires channeling and sublimation, as does any other desire to commit a sin.