Does this mean that Europe and the US can never see eye to eye when a GOP president is elected?
No foreign government can truly see eye to eye with the US when a GOP president is elected, but that has little to do with any discussion of whether Europe is Democratic by nature. (and, no, the various tribes of white people are no more or less Democratic, by nature, than the various tribes of non-white peoples. or democratic, for that matter.)
The GOP's definining characteristic has always been, since its first national convention in 1856, nationalism. Not conservatism, not liberalism. Whether the GOP is the most conservative or the most liberal party around depends largely on what external and internal circumstances, usually beyond the control of any party, dictate. But they are always the most nationalistic party. Sometimes we decide that nationalism is what we need. Sometimes we decide they're not what we need. And the American people have been pretty good, but not perfect, at decided when we want the nationalists in power and when we don't. But when we do decide that we want nationalists, I can assure you it's not during the times when we're seeking ecumenicalism. This would not only be true of the US, by the way. It is my guess that any country would most likely elect members of its most nationalistic party during those times when it deems appropriate, and it is precisely those times when the other governing bodies of the world will meet with the most intransigence. This is neither good nor bad. There will be times when other countries may benefit from Republican administrations, and at any given time some countries will benefit from one US party over the other. But that is generally an unintended effect. But whether or not anyone besides the US benefits when Americans decide that it's time to empower our most nationalistic faction, it will always be the case that the dominant language of American statecraft will serve largely to unify a majority of Americans and alienate a majority of the world outside the US. Sometimes that's the appropriate course, sometimes it's not. And I think we are pretty good at deciding which times are which.