European here too. I thought another problem with closing Guantanamo was what to do with the prisoners.
If they closed Gitmo and brought the detainees to the US for trial, almost all would have to be released.
I'm not necessarily saying that is a bad things, surely some (or even all) do not deserve to be held and should be released, but all it takes is one coming back and committing a terrorist act (could you even blame them after being illegally held and tortured for years?) for the worst PR nightmare a politician could even imagine rains down on them.