My name isXXXXX'm a licensed Texas attorney. I am honestly sorry for your son's circumstances, very much appreciate your patronage, and am glad to try and help out. Please tell him "thank you" from me for his military service on behalf of our nation!
Accordingly, I'm pleased to say that the law is indeed on his side here. Here's how this works. It's a matter of common law (cases) as opposed to statutes. Tort law, and more specifically premises liability law, provides that your son was in the legal capacity of what is known as a business invitee. In other words, he wasn't trespassing (i.e. burglar), and he wasn't there as a matter of right (i.e. meter reader), he was there as a paying patron. That means the business owed him a duty, and frankly the typical outcome would have been for him -- not the business -- to have sought financial compensation for the incident (fall). So, to answer your questions, yes the law does place responsibility upon such an alcohol server (called dram shop liability), and yes their liability insurance would be the appropriate place to turn for reimbursement through filing a proper claim -- not buy bullying your son into paying up. And no, nothing you've described raises the slightest hint of validity in the assertion that he would have gone to jail had he refused to pay. In fact, the scenario you've outlined raises serious questions of their conduct -- not his -- in terms of the issue of extortion and taking advantage of an impaired patron.
If you have a follow-up question or need clarification, please just say the word by using "reply" to reach me.
I truly hope all works out for you and your family.