Can you target your own permanent with a spell that has protection if you control it?

Protection from a quality means the protected permanent cannot be targeted by sources with that quality, period. This applies whether the source is controlled by an opponent or by you. The rule is symmetrical — it does not check who controls the targeting spell, only whether the spell has the protected-against quality.

The four things protection prevents are captured in CR 702.16b: a permanent with protection cannot be Damaged, Enchanted/Equipped/Fortified, Blocked, or Targeted by anything with the relevant quality. Targeting is listed explicitly, with no exception for the permanent's controller.

So if you control a creature with Protection from Red, and you cast a red instant that would target a creature, you cannot legally choose your own protected creature as a target. It is simply not a legal target for that spell, just as it wouldn't be for your opponent's red spell.

Concrete example: You control a White Knight (Protection from Black). You cast Defile (a black instant that targets a creature). Even though you control White Knight, you cannot target it with Defile — protection from black prevents it from being targeted by black sources regardless of who owns them.

Note that protection only stops targeting; some effects don't target at all (e.g., "all creatures get -1/-1") and can still affect a protected permanent. But any spell or ability that uses the word "target" and has the matching quality is blocked by protection for everyone, including you.