There are 2 issues here. The 1st is the school, could they be found negligent for your sons fall? To be negligent, then they caused his fall, or did not properly watch him, but that is hard to prove and most likely a 4 year old runs or horses around with another 4 year old that could have caused his (own) fall. If school is negligent, then would owe for the medical bills and (some) pain and suffering.
The 2nd is malpractice which in your case caused him to sustain the diarrhea, peeling skin and the eyes shutting and subsequent possible toxic syndrome. So that would mean that the cut was not cleaned properly thus the toxins caused this.
But the biggest problem you would have had after coming home from the hospital, is that they (always) on the discharge papers will state if the following occur and (will list these symptoms) then come back to the hospital or to your family doctor asap.
Now you need to keep the wound dry and also make sure that the 4 year old does (not) try and scratch it or the scab, which most likely happened, thus the infection started. Kids don’t usually wash their hands (unless) told to do so, so your child if he/she scratched it, could have caused germs to enter the wound.
So, only (if) you could have proved that the hospital did in fact (not) clean the wound properly and caused the wound to become infected, then you would have malpractice on the hospital/doctor and could sue them for this.
So find out (exactly) how the fall happened, which guessing the school would have filed an incident report of this. Then (after) your child is treated and ok, then make an appointment with a malpractice lawyer to see (if) any malpractice or not. But before you see the lawyer, then ask for all the medical reports from the hospital for the lawyer to review. But malpractice is hard to prove, since you have to document that what the doctor/hospital did was (not) reasonable or expected or errors that could have been prevented.