I don't think I could ever stop someone and say, "No, no, my friend. It's *Doctor* to you."
I think if you're meeting a new patient, it's necessary to introduce yourself as a doctor, so the patient can register the information and define the relationship a bit better from the get go. I do think it's important to maintain professionalism in situations where it's necessary (and I think that's a clinical judgement issue--difficult patients, confrontational situations, psych, pediatrics with the patient's parents).
In life, though? Nah.
At a restaurant once, I saw a guy pay for dinner and the waitress came back with his credit card and said, "Thanks for coming in, Mr. Headupa$$." And he was like, "That's Doctor Headupa$$."
What's the point of THAT, I ask?