I worked in schools for many years until recently, we were all properly Frist Aid trained and I was in charge of accidents and our school First Aid room as part of pastoral care.
You are right in that this was not handled correctly, however you would get best results by being calm and professional when you deal with this.
Accidents do happen and children - young and teenage - do get fractures and broken bones from falls or sports or whatever, so that in itself is rather unfortunate but not unexceptional.
The school should have rung the responsible parent as soon as possible....there might be a delay while they were assessing the damage and dealing with the immediate effects.
If the school does not have a trained doctor they are not allowed to "diagnose" as such, so they wouldn't be able to tell you definitely whether a breakage had happened. But anyone First Aid trained would be assessing the amount of distress/pain, how the child was holding themselves or their limb, and whether any limb was at an odd angle or whatever. It is acceptable to ask a child to wiggle their fingers to see if they are able to (if not it would indicate something quite serious) but not acceptable to 'manipulate an arm or joint' because of the possibility of a breakage. Make sure you know exactly what happened when you say "manipulation" before you make a complaint.
They should be making the child comfortable (sitting down, putting a pillow or cushion under the limb to provide support, child under observation somewhere in case they suddenly fainted) and contacting parents. It would be perfectly acceptable to say to a parent, "he's holding his arm oddly, he's complaining of severe pain, I'm worried in case of a breakage or fracture so we're recommending you get him checked out" - this is quite different from 'diagnosing' a breakage. The school would not expect to call an ambulance in this case as - nasty as a break is - it is not immediately life threatening, and the child can quite easily walk out to parent's car etc.
I would suggest going to the school for the full details, you can ask for a copy of the accident report and take it from there. If they have mishandled it, certainly make a complaint but do it calmly with the full facts at your fingertips first. You will also want to state the outcome you expect from your complaint which would, in this case, be that the school overhauls their First Aid practices and procedures.