Yes and no. He was born fro a forced regeneration that made him a warrior because the eighth doctor believed that there was no more need for a doctor. So practically he stopped being the doctor.
John Hurt's Doctor said at the end of "The Day of the Doctor" "...For this moment, I'm the Doctor again..." so the answer is Yes & No.
The War Doctor is the Doctor, even though he and his successors refused to accept it. John Hurt himself is not the Doctor; he is an actor.