My child who was delusional and having hallucinations was "inappropriately restrained by security at hospital" I was later informed by the hospital that the security guard actually punched my child in the face and was later fired. What should I do? Do I have just cause to sue?

You haven't really given enough information to formulate an answer.

How old was your child? Was your "child" a danger to the staff of the hospital?

"Punching" is an inappropriate action. However, I wasn't there to see how hallucinatory your "child" was and what dangers he presented to the staff of the hospital.

Sue for what? Was your child permanently damaged in some way? Sounds like the security guard was appropriately disciplined.

You asked if you have "just cause". I would say no. You can file suit but I doubt if you would win.