"Gay voice" is an accent, and accents are learned, period. Any sane linguist would tell you that. Now, learning accents is mostly subconscious, but it can also be conscious, as in the case of actors or people who decide to change their native accent for various reasons.

If you listen to what we call "gay voice," it's a mixture of Valley Girl California and African-American Vernacular English, with some likely additions from the speech patterns of female Golden Era movie stars, like Bette Davis, Judy Garland, or Rosalind Russell. Stage voices feature those peculiar exaggerations. It's possible that a subculture of gay men in metropolitan areas acquired those vocal mannerisms from theater or film, aping their favorite stars.

I know plenty of gay men who don't speak with the gay accent at all, so, to me, the idea that it's innate is absurd/ridiculous.