Ironically, without importing this very american view of "black" and "white", people in Europe wouldn't even think in these kind of categories. There was no "solidarity among whites" before that, the peoples of Europe were divided by nationality and more granular ethnicity than just "white". By imposing this retarded focus on being white, it only served to create a feeling of solidarity for other light-skinned people in the first place.
Those who dislike other races won't use "European", though, as "European" as a self-identification is mostly used by the more left-wing cosmopolitan people here as a sign of anti-nationalism.
Also, removing "white" as a category is a bit more complicated (as malleable sociology bollockeries usually are, they can be made to fit any given situation forever and ever). If people started using "european" instead of "white", then "european" would basically be the new "white". The point is not the word, but the concept of the category by itself. Funny enough, the current american usage of "black" isn't even really applicable in Europe due to a very different history, so they just use "people of colour" as a stand-in for "everyone of non-native ethnicity". Removing "white" as a category seeks to eradicate the concept of native ethnicity at all. By extension it would also seek to eradicate the concept of non-native ethnicity and thus ethnicity in general, but by focusing so much on ethnicities and the categories they're just making people MORE aware of ethnicities, especially when proposing all sorts of affirmative action. This, quite ironically, works against their goals in my opinion, and just leads to further divide among the people.