The Church's official stance is men with "homosexual tendencies" are allowed to serve in the priesthood, as long as they "practise homosexuality, present deep-seated homosexual tendencies or support the so-called 'gay culture.'" (Taken from "Instruction Concerning the Criteria for the Discernment of Vocations (2005)." Basically, it's like "Don't Ask, Don't Tell," plus the priests are not allowed to have sex with men (but then, they're not allowed to have sex with women).
That isn't to say that this is okay. It would be better if openly gay men were allowed to serve; to acknowledge, "Yes, I am gay, but I choose not to break my vow of celibacy." Because that's what it's really about--not whether the priest is choosing to not have sex with women, men, or both.
Like the rest of the world and society, the Catholic Church is also evolving. It may take a while but they will eventually have gay priests in the church, sooner than you think. I venture to say that there are already gay priests in the church who have been forced to stay silent. Mankind will move forward quickly once we are honest with ourselves about human sexuality.
The Catholic church will eventually allow gay priests to serve in the church. Society is moving to become more inclusive, and the Catholic church will have to do so, or risk losing more members. In particular, the church needs more young members to replenish its aging population. This demographic is more likely to accept gay members and clergy.
The Catholic Church will not allow gay priests to serve in the church as these will be contravening what the scriptures says. Also if it is to happen, it is bound to bring division in the church splitting it into two or loose the African congregation who are opposed to gay marriages leave alone allowing accommodating a gay priest. I don't think Vatican can risk that.
The Catholic church is a long and storied institution that has many ingrained traditions. One of these, the opposition to birth control, has only recently started to reverse. I think that means that other traditions, such as the barring of gay priests from serving, will take even longer to become the new norm.