Yes.
My son is Jack, and his birth certificate reads Henry John (after my grandfather).
A friend of mine at high school was named Jack only - not John.
This is an education for me, as I had always assumed that most people called John would have “Jonathan” on their birth certificate.
So Jonathan = John = Jack seems perfectly valid for me.
My brother’s daughter is named Sydney. They call her Beaner. (Sydney > Sydney Beans > Beaner.)
I’d say Jonathan > Jack is pretty tame.
Over time I have learned that logic and names are not related.
Spellings, pronunciations, diminutives and anything associated with names are all “up for grabs.”
Jack could be an alternative to anything you want to name. And you don’t have to spell it J-A-C-K or to pronounce it to rhyme with “crack.”
Be your own boss in the matter and dare others to dispute your rights.
To paraphrase an old joke of questionable taste: “it’s my mouth and I’ll haul coal in it if I want to.”
I don’t think many Jonathans are called “Jack,” so I would consider that different, but that’s all. It’s pretty common for people to go by some name that’s not on the birth certificate, so who cares. . .
sure why not it will be a welcome change after the nick name pukepoopmachine
John is a short for for Jonathan.
Thus you could call Jonathan, John
Jack is a nickname for John
Thus you could use it
You’re just taking the long way or scenic route 
Actually, it’s “Jon” that’s short for Jonathan. IIUC, “Jon” is almost always short for “Jonathan,” while “John” is hardly ever short for anything else.
It’s never been clear to me why a “John” would need a nickname, or if so, why “Jack” should be a nickname for “John.”
But, if “Jack” can be a nickname for Clive, I don’t see why it can’t be a nickname for just about anybody.
All this, of course, is assuming the OP’s son is okay with being called Jack. He ought to have veto power over what he’s called.
Just as long as his last name isn’t Schitt. 
Be sure you tell your boy about the part where you asked for advice about his nickname on this quaint old relic we used to call an “Internet Message Board” so that he can work that into the story.
IIRC, Margaret gets shortened to Maggie, which gets baby-mispronounced as Peggy.
Yeah! Back in aught-ten.
My son’s name is simply his name on paper. I’ll call him whatever I like, and other members of my family will do the same. If my father were still alive, he would call my son George, and nothing could be said or done to stop him. It’s tradition.
That being said, my son’s name is Jackson, and his “official” nickname is Jax. Personally, I love nearly all variations of Jack, so I’m a proponent, anyway.
At one time, John was the most popular given name in England. (cite) Having at least one alternative or nickname must have seemed a good idea.
I think Jack is an American invention, actually.
No, Jack is one of the oldest names. Most fairy tales and nursery rhymes use Jack because of its common association with the average man.
Joe may be of uniquely American origin, though. Just guessing.
Double Post
I love this thread! Jessica -> Ruth is killing me. Jack for Henry? Brilliant!
And, really, Jack for Jonathan is kind of growing on me, too. My other son is Alex (not Alexander; I took into account the crayons and stubby fingers with that one ) James but he won’t let me call him AJ. 
Jonathan is only 3 weeks old today so he doesn’t really get any say yet.
No, it’s not.
JON is short for Jonathan.
JOHN is a full name in its own right.
John is from the Hebrew Yochanan.
Jonathan is from Yonatan.
Two different names, two different derivations.