Surely that would simply have it count as part of the Papal States?
Although, the idea of the Pope as a sponsor is certainly an interesting idea. The poor come to Rome, and Rome pays to send them to the New World, complete with supplies, navigator, a map - and to spread Christendom.
I could see that working and creating lots of independent statelets in the Lesser Antilles. Twinned leadership of a secular head and a bishop/vicar/church rep, that could form the heart of a confederation/federation tied together by the Church, with larger expeditions eventually leading to the Greater Antilles.
The way to have it happen would be mad though - an alt-Treaty of Tordesillas where the Pope says "It is for neither, it is for Christian Pilgrims". It could be interesting to see monarchs denied, but minor nobles who don't inherit to fund, or jointly fund with other minor nobles, colonisation missions since, well, at home they aren't going to get land. Their fathers might even help fund them as it bring stability to the estate.