perhaps "his doctor was John's book" would give the clearer impression

Both sentences can be improved for clarity and naturalness. The second sentence is grammatically correct but could still be ambiguous. Here are the revised versions: "He has the doctor's book, John's." "He has a doctor's book, John's." However, if the original intention was to indicate that John is the owner of both items (a doctor and a book), the sentences should be rephrased for clarity: "He has John's book and a doctor." "He has John's book and the doctor." Context is important to determine the exact meaning, but these revisions aim to make the sentences clearer and more natural.