The Apostolic Palace is much more situational than the United Nations. The chance must arise before there are any decisions to be made.
In order to win a Diplomatic victory from it, the religion needs to have been spread to at least one city in every civ in the game. (so that everyone gets a say, even if it's only worth a few votes)
In the Civilopedia, there's a section for Beyond the Sword Concepts for the Apostolic Palace. All of it's functions are listed there.