The short answer is yes, he will be fine. There is no point torturing yourself (and him) over this. Let it be. In fact, if you relax your worry there is a bit more of a chance that he'll expand his menu a bit. One thing you might do is arrange two evaluations: one by a Speech/Language clinician to evaluate his oral motor muscles and swallowing, the second evaluation by an Occupational Therapist to evaluate for a possible sensory integration problem. If these evaluations don't show any problem, please stop worrying about this situation.
I know you posted this for the doctor, but I thought I'd jump in since this sounds so familiar.
Until my brother was 11 or so he basically only ate peanut butter, bacon, apples, cheese pizza, french fries, raw carrots and white bread. Getting him to try new foods was a nightmare. Eating was a chore for him, he did not enjoy it and basically only ate what he had to survive. When we went to a restaurant, he carried in a jar of peanut butter and ordered a glass of milk. You probably don't want to hear that it can last that long... but he's a healthy 48 year old now. (still a picky eater though!)