Does an aura attached to a permanent get destroyed when the permanent gains protection from the aura's color?

Protection from a quality means, among other things, that the protected permanent cannot be enchanted by sources of that quality (CR 702.16b, the 'E' in DEBT). If an Aura is already attached and the enchanted permanent later gains protection from the Aura's color, the Aura's attachment immediately becomes illegal.

State-based actions are checked continuously, and CR 704.5n specifically states: if an Aura is attached to an object or player it cannot legally enchant, it is put into its owner's graveyard. This check happens the next time state-based actions are checked after the protection is gained — no player receives priority between gaining protection and the Aura being moved.

Note that this is distinct from a spell or ability trying to attach the Aura; protection would stop that too. But even a pre-existing attachment is ended by protection via the state-based action rule, not a 'destroy' effect — so regeneration and indestructibility do not save the Aura.

Example: Your opponent's Grizzly Bears (a green creature) has Pacifism (a white Aura) attached to it. You cast Brave the Elements targeting your white creatures — wait, wrong scenario. Instead: Pacifism is on your opponent's creature; the creature gains protection from white (e.g., via Apostle's Blessing). State-based actions are checked, Pacifism is no longer a legal attachment, and it goes directly to its owner's graveyard.

Remember: the Aura is not 'destroyed' in the rules sense (so indestructible doesn't help), it is simply moved to the graveyard by state-based action per CR 704.5n.