I answered a similar question with sources here.

First, it's important to note that the shift from apprenticeship to formal schooling for physicians in the United States was a gradual one. In the 18th and 19th century, for instance, physicians-in-training would combine both university courses and apprenticeships to learn medicine. In some ways, this system remains true today, as medical students in the USA and Canada complete two years of preclinical work in the basic sciences and then two years in clerkship rotations, where they are taught about clinical practice in hospitals by residents and attending physicians.

It wasn't until the Flexner report of the early 20th century that medical schooling was becoming more formalized, and the recommendation that it be post-graduate (i.e. that students hold a BA/BS before entering) was ever made. Part of the shift to formalized schooling was that as medicine incorporated the biosciences into training for future physicians, students needed access to educational laboratories, classrooms, and lecture halls that only universities could support. The adoption of science into medical practice was also something of a political move, as a means of allying the profession with the 'hard' sciences which were growing in prominence and respect and therefore conferred more legitimacy onto physicians (See Rothstein's "American Medical Schools and the Practice of Medicine" and "American Physicians in the 19th century." See also Starr "The Social Transformation of American Medicine.")

I am not an expert on European medical education, although I would refer you to Bonner's "Becoming a Physician: Medical Education in Great Britain, France, Germany, and the United States 1750-1945." I can say that European universities have a lengthy history of offering coursework in medicine at the bachelor's degree level. In the Renaissance and earlier, medicine, law, and theology were the three main 'tracks' of coursework at all universities, and at that time there were fewer distinctions made between BA/BS and graduate training than there are today. This may have carried over into the existing model for training at European universities that we see now. You should also note, however, that there are some American universities that offer accelerated BS/MD tracks where students complete their medical training in a similar, condensed time frame as European students.