The incidence rate of HIV infection is the same in the US as it is in France, Span, or Italy though higher than the Scandinavian countries. I couldn't easily find data for the current rate of new infections, but its not immediately obvious to me that access to health care is the major driving factor.
I don't know enough to comment on European countries, but I wouldn't be surprised if access was a big problem there as well. The American healthcare system is pretty bad, but it's not the only bad system in existence. There may be other linked factors like stigma at play as well.
https://en.wikipedia.org/wiki/List_of_countries_by_HIV/AIDS_...