Doctors aren't trained in education. They're trained in medicine. Ignore the doctor. Actually, most psychologists aren't trained in giftedness either. They're usually trained in things like depression, anxiety, and adhd.
He'll be fine. Read volume 1 of this, and if you have time, read the sections of vol 2 that apply to you. (They're both basically the same book, but vol 2 has a ton more details and statistics. Vol 1 is just an overview.)
IRPA - A Nation Deceived - Get Report
Welcome to the board, by the way!

(Should've started with that little detail, huh?

) Is there a way that he could get some exercise first thing in the morning to calm him down a bit? Can you role-play with him by cutting him off mid-sentence and stealing his ideas so he knows how the other kids feel? The problem with grade skipping isn't that it's a bad fit academically (usually... I'll comment on that in a sec). It's that some of the kids in his class will be 2 years older than him because of either failure, where their birthday falls on a calendar, or "redshirting". Is there a possibility for him to have more challenging work, either by a pull-out group, gifted enrichment in the classroom, or switching to a gifted school?
Grade skipping isn't always the answer. Kids who are accelerated will do well with a grade skip, because it places them in the grade they should be in. Kids who are truly gifted (depending on the variety of giftedness) may one day become bored again. The grade skip is a short-lived solution, because gifted kids simply catch on faster than average kids. They think deeper and more quickly than their peers, so eventually, they could master the information in the higher grade level long before the year is over.
All of this is discussed in the link I gave you. Grade skipping was probably the best choice for now, but you may have to adjust that choice again later in some way.