Why exactly would Catholic monks lead the first expeditions? Prior to Columbus's journey there's no good reason for Catholic monks to cross the Atlantic, and afterwards the secular authorities have every reason to explore, invade, and colonize these lands. Furthermore, I question the idea that the Catholic Church was all about peaceful conversion. The Church had a long history of supporting military ventures to conquer non-Christians, and actively encouraged the sort of behavior that the Conquistadors engaged in. For example, in 1452 the Pope granted issued a bull (Dum Diversas) which granted Portugal "full and free permission to invade, search out, capture, and subjugate the Saracens and pagans and any other unbelievers and enemies of Christ wherever they may be, as well as their kingdoms, duchies, counties, principalities, and other property [...] and to reduce their persons into perpetual servitude." This right was confirmed in multiple other bulls made prior to the discovery of the Americas. And that's not even getting into the Crusades and other religious wars the Church authorized. When the Pope gave Spain the right to the New World in 1493 he was following a long tradition. There are individual religious orders that supported peaceful conversion, but they don't have the power to stop the Spanish invasions.