The game comes first and foremost. In some situations it really wouldnt matter. An atypical brown and bloom filled shooter, it wouldnt make a difference because it has zero impact on the mechanics and the narrative in most shooters is so small that it would barely have enough room to be expounded upon in a 7 hour game. (besides who plays shooters for the story?)
Now a more narrative driven game like an RPG, that Im a little more on the fence about. Then it depends how it is done. If its a non bioware game where sexuallity is seldom an issue in the first place, and your not constantly being beaten over the head with it, then as far as I am concerned, its fine. Now if its something that keeps coming up over and over and over again, then no. It would become entirely too distracting and end up disolving the narrative down to that facet of this virtual persons existence.
Arguably this can already occur. Fable is the perfect example. You are free to screw whatever you like in fable From begining to end if you want your char to be gay, its gay. Thing of it is, games are extremely progressive when it comes to this subject. You see more liberal and tolerant views when it comes to homosexuallity in games. Almost to the point of being too progressive, Or being progressive just for the sake of being progressive. If you look at games being representative of our culture, then already the attention homosexuallity gets in games is either right on par or even just a little bit over considering that homosexuallity makes up roughly 10-20% of the global population. You look at the level of games where homosexuallity is touched apon and it prolly hovers at a similar percentage.