It usually means (in the context of a signature) that he has a diploma from somewhere. Usually, the place where it is from would be identified, though.
Oh, I’ve seen something like this – it’s more often Diplomate (which your wife or you may have misread as Diplomat) – and it means that he’s an accredited graduate of a school, probably in Europe, which grants “diplomas” rather than baccalaureate degrees.
Maybe it’s like that doctor on “Friends” who tells people that he likes Fonzie.
OK, seriously – this is very much rank speculation on my part, but maybe he really is a diplomat. (Or, it sounds more likely, the spouse of a consular official or something like that.) As such he is probably immune from either criminal prosecution or civil lawsuits, even for acts he does that have nothing to do with the business of the diplomatic mission with which he is associated. (See, IIRC the document’s name, The Treaty of Vienna on Consular and Diplomatic Relations of 1963.) As a patient, I would definitely want to know if my doctor had no fear of a malpractice judgment being rendered against him, as this might make some doctors less careful in their procedures.
This also indicates specialty board certification. If, for example, he indicates that he is a Diplomate of the American Board of Family Practice, this means that he has passed the certifying exam given by that Board. The certification exam for that paticular Board has to be taken again every 7 years.