I'm guessing you either misheard or misinterpreted that statement (the alternative being that your professor was woefully incompetent). To take early Colonial Mexico as the example, while there were heresy trials and even religious executions, these were so few in number as to be demographically insignificant.

The real population collapse came from Afro-Eurasian diseases (and possibly emerging indigenous disease) introduced to the immunologically naive populace of the Americas. Section 2.b. on this page discusses some of the epidemics and populations estimates.

The emerging epidemics were coupled with, particularly in the early encomienda system, incredibly harsh labor obligations which were fundamentally no different from slavery. Moreover, the labor obligations compounded the effects of the epidemics by creating a population already over-stressed and underfed, while putting them in conditions where communicable diseases could easily spread.

This is, of course, aside from the of wanton slaughter that also took place throughout the Americas, both sporadically and as a matter of official policy.