But let's suppose the visit of the Duke of Cambridge to Canada (hypothetically) makes him and monarchism become very popular there, si that a large part of the population would like him to become governor-general, is there any real constitutional impediment for him, if the Canadian Prime-Minister would ask the Queen to appoint him to the position? Or would it just not happen, because a royal governor-general would antagonize the republican part of the population, and thus not be a good candidate for the job?
It just wouldn't happen. Since the appointment of Vincent Massey as GG in the '40s, you have to be a Canadian citizen in order to be GG - members of the Royal family do not count in this sense, particularly if one of the members becomes a future monarch. So there would be major constitutional impediments right there, even if certain members are popular. No republicanism needed, as the republican movement in Canada is very small and inconsequential anyway.
Not just that, in Canada one alternates between anglophones and francophones for GG, and in this case Canada's already in its anglophone phase right now.