Say that Spain sells Aruba to the Papal States in the 16th century (after Lepanto, maybe). Spain gets some cash, the Pope gets a temporal foothold in the New World. Assuming the Papal States manage to keep Aruba, how does this odd little colony develop?
Were there slaves on Aruba? There might be some moral/ethical questions there.
Generally the extra-Italian Papal holdings were frequently occupied by the Papal court and bureaucracy (i.e. Avignon). The Papal States would likely reject the offer unless Aruba was economically and ecclesiastically profitable.
Honestly if the Papal State end up with Caribbian colony, it will be treated like all other colonies there, a money maschine for the owner, so the effect will be that the Papal States have a little more money. Later it will be more interesting, what will the effect be when/if Italy conquer the Papal State, will the colony stay part of the Vatican and what effect will that have?
Pope moves to the New World? If he has enough Swiss Guard and other soldiers there, he can think that the island is his safest location.