There are real indications that homosexuality is something folks are born with. Our sexuality sits on a spectrum, as does many other behaviours we have that range from no-so-much to very-much-indeed (sexually we measure with the famous Kinsey Scale that rates our sexual history or episodes of sexual behaviour and inclination). The sub-set (for want of a better word) of gays who exhibit the effeminate "gay" voice often share other traits as well.
There is a way of walking and moving, an over-all physicality that has feminine traits. I think this is where the term "gaydar" comes in. People read the non-verbal cues and make judgements all the time. A genetic male sometimes demonstrates femininity in his way of running, moving, speaking or gesturing. There is perceived disparity between the x-chromosone weighting his overt appearance and the more subtle expressions guided by a genetic influence given by what ever it is we inherit that says "I am gay and that's that". BTW I am not too sure this "gay voice" is a litmus test for homosexual inclination. My folks have had a neighbour for many years. He has a large family and is a lovely man, and he has a "faggy voice" along with all the rest of it that sets off my gaydar. But there has never been any actual indication other than his manner of speaking and movement of any homosexual tendencies, so it may just be a coincidence that so many gay people have that collection of tendencies. The flip side is the butch lesbian. I'm not too sure the gruff, robust manner of many gay women is put on. I'm betting it also is part of some as yet not identified genetic component in their makeup.
There are other things less overt physically in many gays that exist as a way of being. Things such as greater empathy, creativity, expressiveness, and some other elements that generally are accepted as being on the feminine end of the human behaviour spectrum.
The "gay voice" is no more an accent or a conscious attribute than a lisp is, or that wonderful thing that James Earl Jones, Morgan Freeman and Kathleen Turner have going on. It's just another manifestation of your inner, born self expressing naturally as you communicate with others.
btw Zeraph - that was one funny visual you gave me with the ring thing you riffed on