The natives in the Indies plainly weren't going to fall into the category of Old Christians. Were they viewed as suspiciously as conversos?
How does this square with the Inquisition's dealing with them? As I understand it, the Inquisition's jurisdiction explicitly excluded native populations because the Catholic faith was not yet mature with them, but isn't it odd that the very instrument for religious orthodoxy wasn't used to ensure an orthodox development of the faith among the natives?
You may be interested in one of my posts here (from an old account) and here.
The indios of Peru were essentially a "blank slate" compared to their fellow Muslim or Jewish converts. They were open vessels ready for Christianity, with some of the foundations of civilized "Spanishness" already there: bureaucratic government, orthogonal planned cities, etc. There was no need for the Inquisition to target them; they couldn't exactly be Jews or Muslims faking their Catholicism. The native beliefs didn't spoil sangre in the same way that other Abrahamic faiths could. Of course, other measures were taken to ensure local's faith, but these were executed at a more local level, entirely disconnected from the Inquisition.