It depends on where, because for the most part we are talking more about rules than laws.Speaking generally, by the late 1970s hospitals gradually came into compliance with court decisions and administrative rulings. One complication has always been that the government demands racial stastics from schools, hospitals, employers.etc., but usually forbids the relevant questions being asked.(!)

In your example, you would have to contact the state's bureau of vital stastics and ask specific questions like:"Is race recorded on a birth certificate?" and, if yes, "Who makes that determination?" My guess is that the answer is no. Hell, they don't even ask if the parents are married, in most jurisdictions.

I'm certain that any hospital would be in serious trouble if they attempted to indicate what they 'assumed' an infant's race was on a birth certificate. There have been too many court cases and much litigation. Most municipalities have their own forms, so there are thousands of variants. The few recent ones I've seen don't indicate 'race'at all, although older ones, like my parents, do. Concerning the law you mention regarding some degree of black ancestry requiring birth certificates that reflect that fact; such things ceased to exist generations ago, long before that class you took, probably taught by an instructor whose righteous zeal may have led him to bend the facts a bit, or present some unhappy history as being contemporary.