Does an enchantment aura fall off when its enchanted permanent gains protection from enchantments?

Protection from a quality means the protected permanent cannot be enchanted by sources with that quality (among other things). This is part of the DEBT acronym: protection prevents Damage, Enchanting/Equipping, Blocking, and Targeting by the specified quality (CR 702.16b).

When a permanent gains protection from enchantments, the game immediately checks all Auras attached to it. Any Aura that is an enchantment — which all Auras are — is now illegally attached. State-based actions (CR 704.5n) then move any such Aura to its owner's graveyard the next time state-based actions are checked, which happens before any player gets priority.

This happens automatically without any player choosing to do anything. The Aura simply 'falls off' (is put into the graveyard) as a state-based action the moment the game checks SBAs after the protection is granted.

Example: Your opponent controls a Darksteel Mutation (an Aura) enchanting your creature. You cast Apostle's Blessing giving that creature protection from enchantments. Before anyone receives priority, state-based actions check: Darksteel Mutation is an enchantment attached to a creature with protection from enchantments, so it is immediately put into your opponent's graveyard.

Note: if the Aura somehow is not itself an enchantment (a very unusual edge case that doesn't exist in normal Magic), protection from enchantments would not remove it. But all Auras have the enchantment type, so in practice the Aura always falls off.