Not really. Did you know that wine is mostly considered not to be vegan because the white of eggs is used to clear the suspension of cloudy stuff that makes wine look unappealing? Same goes for champagne: not vegan! That is also the reason why in many places where wine is made there are lots of traditional sweets based on egg yolk, because that is what is left from the clearing process. Yemas de Ávila (also called Yemas de Santa Teresa, yemas meaning yolks) for instance, from Spain. Yummy, but extremely sweet.
Of course there are also vegan wines, they use clay or other non animal stuff to clear the it, but that is (in my understanding) a rather recent development and still not very widely used.
So again: no, egg white does not alter the taste of wine because it is already there and you don’t notice it.