My quick look to find out Arizona law indicates that some corporal punishment is allowed in Arizona. It seems that whatever happened might not have been legal. If it was not, then hitting your child was battery and subjects the teacher to a claim for damages. The problem may be that the amount that could be recoevered is so small that it would not interest a lawyer in taking the case. You might be able to file a lawsuit in small claims court, but how you would prove damages without a lawyer would be a problem.
You should probably be grateful that the school let the teacher go and that the reason will be on the tesacher's record.
If you want to find a way to pursue this, you might be best looking for a young lawyer who will take a case that will not result in much of a fee, but who has the time to handle it anyway.
DISCLAIMER: I have not agreed to be your lawyer or to give you any legal advice. This response is for information purposes only and is not legal advice. For legal advice you need to find a lawyer who will agree to represent you in this matter.