To white out a dungeon ...

cabbageub17_ESO

Does it need to be run on I and II? And if so is there a way to tell if you've done it on one or the other?

Humble48

I think the dungeon whites out and earns you a skill point once you run it the first time in any difficulty. To tell the difference, I think you could check your achievements for completing it.

DMuehlhausen

I believe it's after you do the quest associated with it. So yes you would probably need to run it on I and II.

AlphaAbsol

You need to beat the final boss in the dungeon (each dungeon if there's I and II). It can be on normal, and you only have defeat the final boss (you don't even have to do the rest)