If you call the character "Doctor Who" then #DoctorWho fans will correct you and say "Actually, the character's name is just The Doctor, not Doctor Who".
Don't listen to them! They're wrong!
- In the credits to the very first episode it says "Dr. Who: William Hartnell"
- In The War Machines, evil computer WOTAN says "Doctor Who is required"
- There's an episode titled "Doctor Who and the Silurians"
There you go, the character's name IS Doctor Who! Canonically!