Seeing Spanish conquistadors set up their own mini-states isn't too hard; Cortes was suspected of it, and you had a few people like Gonzalo Pizarro who made halfway decent runs at it. The Crusade bit isn't needed. It's not like the Pope created the original Crusader states, he just told people to go liberate the Holy Land. Creating independent states afterwards (as opposed to returning them to the Byzantines or even creating a Papal enclave in the Levant) was something decided on by the Crusaders themselves.
As for non-Spaniards trying to Crusade? The Spanish were very restrictive on who they allowed into the New World for a host of reasons. Foreigners had great trouble getting in for any reason; foreign would-be Crusaders are obviously right out. A foreign power like England or France could certainly sponsor such an expedition (as indeed, they did at times), but that's going to turn into an English or French colony.