Is it proper for a young PhD to call himself as Dr ?

I received my PhD a couple of months ago in the field of Computer Science. Few days ago a had to go to a hospital due to a stomach ache. A physician examined me (apparently there was nothing wrong with the stomach, just bad food :) ) He then asked my name for the paperwork and I responded "my name is Myfirstname Mylastname, Dr. Myfirstname Mylastname". "Are you a doctor??" he responded. I explained him that I am NOT the kind of doctor he is. I noticed signs of contempt after that. I did not referred to myself as Dr out of arrogance, but because I thought it is completely natural for me now. Or it is not?