I'm not sure you'd even need to bring religion into it -- in lots of ancient societies people who fell irrevocably into debt were either made their creditors' slaves, and I could just see the Southern elite trying to introduce the same thing into their own society.
As for the Spanish vs. US difference, I think this might partly be because the early Spanish Empire was mostly conquered by single young men who went over without any women, meaning that when they decided to settle down their only real option was to marry native women; and, when they did so, they naturally didn't want their own children to be condemned to a life of racial oppression. The original colonists in the Thirteen Colonies, OTOH, had many more women in their numbers, meaning that everybody could marry fellow-colonists without having to get involved with the natives. (You could compare this with the situation in British India, where mixed-races marriages became much less socially acceptable in the nineteenth century as more European women started to come over.)
Interestingly enough, in Rome freed slaves became full Roman citizens upon manumission, although that might have had something to do with the fact that, at least in the early days, slaves would mostly be captives taken from the neighbouring city-states, who therefore came from a similar cultural and racial background to the Romans themselves (plus, Rome's original population was an amalgamation of two or more different tribes, so concerns over racial purity never really loomed large in the Roman psyche). In the Deep South, OTOH, the slaves were mostly taken from a totally different culture thousands of miles away, and looked rather different to their enslavers.