Upvote:2
According to a book chapter on physician migration, "30% of physicians who entered residency training [in the United States] between 1950 and 1976 were foreign born and foreign trained, a phenomenon that was unprecedented over the previous century. The comparative figure for Canada was even greater at 50%." So clearly there was no shortage of North American hospitals sponsoring residents from foreign medical schools in that period.
EDIT: Regarding visa sponsorship specifically:
From 1952 through 1978, the ECFMG, along with some 225 select educational and scientific institutions, served as visa sponsors for J-1 exchange visitor physicians (βalien physiciansβ) who came to the United States. (source)
After that and until now, the ECFMG is itself the primary sponsor. As for how prospective medical residents find their institutions, the National Resident Matching Program was created for this back in 1952.
It is possible that Indians also went to study medicine in the United States during that time, but I'm not finding any numbers on this. In that case, they would have likely come on student visas sponsored by the medical school.