Unfortunately, I think you are without any remedy in this situation. The league is most likely a private entity or organization, and as such, is free to make their own rules with very few limitations (so, they could, for instance, decide one day that every child with blue eyes can no longer play sports, or only kids taller than 6' could play, and that would be allowed). Constitutional protections, with few limitations, do not apply to private companies or organizations. Therefore, even if you had a case based on something like discrimination due to race, religion, or national origin, it likely would not be successful.
At best, ***** ***** possibly be a case of negligent infliction of emotional distress (NIED). Emotional distress is typically defined as mental distress, mental suffering or mental anguish. It includes all highly unpleasant mental reactions, such as fright, nervousness, grief, anxiety, worry, mortification, shock, humiliation and indignity, as well as physical pain. It can be hard to prove, because what one person considers humiliation, another might consider gentle teasing.
The elements of a prima facie case for the tort of intentional infliction of emotional distress are:
(1) outrageous conduct by the defendant;
(2) the defendant's intention of causing or reckless disregard of the probability of causing emotional distress;
(3) the plaintiff's suffering severe or extreme emotional distress; and
(4) actual and proximate cause of the emotional distress by the defendant's outrageous conduct.
In other words, you would have to prove that being denied the right to play caused your son "severe or extreme" emotional distress and that he suffered damages as a result. Being understandably upset about not being able to play unfortunately isn't compensable damages. If, on the other hand, he was so traumatized that he ended up in a doctor's care, that might be damages. My opinion is that you would have a difficult time getting this in front of a judge.
You could always discuss the matter further with a local personal injury lawyer. Generally, the consultations are free, and if they do take the case, it's usually on a contingency basis, meaning you don't pay anything unless the lawyer recovers.
All that said, the league must have a board that sets its rules and regulations that you could appeal to -especially if your son is right around the cut off point. You could tell them that your son has been playing for years, is skilled, etc. I'm sure it's a safety thing for them -they're likely afraid smaller kids will be hurt -but perhaps knowing that he's not new to the sport and has played with bigger kids before (if he's played with kids taller than him) they would grant an exception.
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