A colleague of mine who has spent much of the past 10 years in Germany moved back to the US recently and says he misses German healthcare. Apparently, he cannot pick the doctor he wants; he has to make sure first that the doctor is certified (or whatever) with his insurance company.
I left the US when I was 24, so my contact with healthcare in the US is both scant (those were my healthy years) and a long time ago. Honestly, I have no idea what healthcare in the US is like, aside from recollections about what my father, a pharmacist, said. But I have been hearing in US media that Americans don't want "socialized medicine" of the kind we have over here because they fear not being able to choose doctors. So it surprises me that you already can't choose your doctor in the US.
I am not sure what the situation is in every European country, but certainly you can choose your doctor in France and Germany. I would be surprised if the situation were different in Spain, the Netherlands, Switzerland, etc., but you have the comments box below if I am wrong.
My understanding is that the US was considering a public option alongside all of the private healthcare, which would have made the US system look a lot like the German system (and a lot unlike the French system). I do not completely agree with my colleague, however, that German healthcare is all that great. Your average German would tell you that we have two types of medicine over here: one for first-class citizens and one for second-class citizens. If you are in the public system, which tries to keep costs down by not paying too much, then the doctors tend to make you wait longer than if you have private healthcare, which tends to pay better, so doctors like it more.
Nonetheless, Americans apparently seem to completely misunderstand both their own healthcare system and what they call "socialized medicine," at least when it comes to the freedom to choose your doctor.