SW Alabaster

You don't need to decide on a paint color until everything is in place. If you decide too soon it is easy to make a mistake. Once everything is installed cast a wide net with paint chips and see what appeals to you. You didn't say whether you want a white, off white, or a different or darker color. It's possible that Alabaster would be too warm, if so and you want a white, look at SW Pure White. BM Classic Gray might be one to consider as well. It's an off white.

I would like to see if the walls can act as a grounding color since the tile turned out to be

a bit washed out "color-wise" because of the variation in the dyeing process.

Does this vanity I purchased dictate the color of the walls? (style, wood tone etc?)

IMO you have enough sort all one color already so why not think a nice watery blue / gren for instance but I agree wait until all the rest is in place and if this is a master bath then what is the decor in the bedroom Watery from SW is what I am thinking but maybe cut a bit for a nice pale color