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I have a card with the following ability:
{T}: Target creature you control gains protection from the color of your choice until end of turn.
If a white card like Pacifism has been attached to the creature, am I able to tap the creature in order to give it protection from white and thus removing the white spell?
Yes, you can do that. In fact, you don't need to do it in response. If at any time the creature with Pacifism on it gets protection from white, the Pacifism will no longer be legally attached to the creature, so the Pacifism will "fall off" and go to the graveyard.
The rules for protection include these:
702.16b A permanent or player with protection can’t be targeted by spells with the stated quality and can’t be targeted by abilities from a source with the stated quality.
702.16c A permanent or player with protection can’t be enchanted by Auras that have the stated quality. Such Auras attached to the permanent or player with protection will be put into their owners’ graveyards as a state-based action. (See rule 704, “State-Based Actions.”)
So, if your opponent casts Pacifism targeting a creature, and you activate the creature's ability, then Pacifism can't target the creature anymore, so it will be countered. If you activate the ability after the Pacifism is attached to the creature, the Pacifism will no longer be legally attached to the creature, so it will be put into its owner's graveyard.
The situation is slightly different if the "pacify" effect in question is a card like Arrest that says that the creature's "activated abilities can't be activated". In that case, once it is attached to the creature, the creature won't be able to activate the ability that grants protection, so in that case you would need to activate the ability in response to the aura spell.
Murgatroid's answer is spot on from a legality point of view. This began as a comment but quickly grew to that answer. The simple answer is that you can do it before or after the aura resolves to keep your creature aggressive, but there are consequences in both directions.
The most common consequence to tapping before it resolves means you can't use that tap ability for something else. It is a common tactic to use your second most effective spell first in an attempt to bait out your opponents countermeasures. Since Pacifism is only relevant during the combat phase, it makes little sense to respond during the first main phase.
You don't know what tricks your opponent might have later in the turn, so you could keep that in your pocket until you need it. For example, after your opponent has declared attackers, you get priority and use your protection to shrug it off and declare your previously-pacified blocker. But if your opponent has another trick up their sleeve, you might want to save that protection ability until you actually needed it. Using it early means you can't use it somewhere else.
Of course, delaying and allowing it to enter the battlefield does have other consequences, although they're usually tactical edge cases, not strategic guidelines. Some cards, like Sage's Reverie care about how many Auras you control. Others like Ajani's Chosen are about Enchantments entering the battlefield. If you allow the Pacify to resolve while your opponent controls a Ajani's Chosen, they will get a 2/2 Cat Token even if you immediately remove the aura post-resolution.
So in general, it is a good idea to allow it to resolve and shrug it off only when you need to, like right before you declare blockers, or on the end step if you haven't found a better use for the protection ability. There are cases, however, where it is tactical to shrug it off in response and never allow it to enter.
{T}: Target creature you control which presumably means they're Pacifying your Wurmcoil Engine, not your Mother of Runes.