I'm not certain what you mean by the "Western Empire's legitimacy". In any case, I'd hardly consider it a comparison. Granted, America was certainly the new big kid in the playground, but there were still millions of people living in the Americas that were loyal to the Crown; that cannot be ignored.
My idea is that given a scenario of say, Justinian I successfully reconquering the territories of the former Western Empire, if a general revolts and proclaims himself Western Emperor, what territories would he need to gain before the title becomes legitimate, at least to historians? If he holds nothing but Britain, then his claim is something of a joke, but if he takes everything except Africa and the Med islands, then I would say he has a good case. Africa and the Med islands are wealthy territories with large populations, but they do not form the core of the Western Empire. Canada and the Carib islands, I would argue, would be much like Africa and the Med islands in the above example.