I haven't seen Doubt, but I understand it deals with some of the sexism in the Catholic Church in addition to the questions about child abuse perpetrated by priests. I just ran across this story from last month (via Skepchick) about a priest, Roy Bourgeois, who is fighting for women to be allowed into the priesthood.
Catholicism has long held some stereotypes about men and women in faith. Bourgeois is attempting to say that women can be spiritual guides just as well as men. He's gotten a great deal of support from the Catholic community (including this letter) and continues to be ignored by the Vatican.
It's not surprising that the Vatican is entrenched in stereotypes or that they're unwilling to consider women for the priesthood, but it does make me wonder what the Church would be like if more women were given higher positions in the Church. Would the Church still have such a strong anti-birth control stance? Would they still block aid to women in the developing world over disagreements about abortion? Perhaps, but it's hard to say what impact a diversity might have in an organization.