Here's my four-penneth and I think it goes something to clarifying Cornell's original suggestion.
Yes, he is called 'the Doctor'. This is what he refers to himself as and what characters within the narrative of the series call him.
However, outside the show - in the real world that us mortals inhabit - he is often referred to as 'Doctor Who' (or sometimes in print, abbreviated to 'Dr. Who' - see World Distributors, TVComic etc...) In the real world, it's done to differentiate between the character of our favourite Time Lord and medical professionals in general. For example, the question "Have you seen the new Doctor?" could refer to the start of a discussion on Peter Capaldi's first appearance as the lead character in the BBCtv series or to the new GP who's just started working at the local practice. It's more clearer what is being talked about if the term "Doctor Who" is used for the former and "the Doctor" is used for the latter - especially when engaging in conversation with non-fans.
There's exceptions to every rule of course. It wouldn't be reality if there weren't. Back in the early days of the series, 'the Doctor' was referred to in script directions as 'Doctor Who'. This was allegedly to differentiate the character from any actual doctor that appeared in the script. It also ended up being used on the lead actor's credit in the closing titles. As such, new members of the production team - step forward Mr. Gerry Davies - were easily confused. The climax of episode one and the start of episode two of The War Machines in 1966 sees characters on screen refer to William Hartnell's character as 'Doctor Who'. There's also the Doctor signing a document a few stories later as 'Dr. W' and using the pseudonym 'Dr. von Wer' (German for 'Dr. Who'). I'm sure there were letters to Junior Points of View and the production office about this... After all, Twitter hadn't been invented then...
To conclude, as far as I am concerned, it's okay to call him 'Doctor Who' (or 'Dr. Who') outside the context of the series narrative. But woe to any professional scribe who writes dialogue within one of their televised scripts, published novels or audio plays that calls him that... without very good reason! When the phrase "Doctor who?" pops up in dialogue in the show, the lead character will always give the speaker a funny look. It's because he doesn't like it one bit...Here's a rare issue of Whovians In Love that sums up the passion of fan opinion...