Well, if you don't know, then you are not in a union. You'd know, because it is a length process.
You'd also know if you had an employment contract stating that you can only be terminated for cause.
What they are describing is that you are an "at will" employee. That means that you can legally be terminated at any time, with or without cause. So, that is what they appear to be telling you and, for a private school in Texas, that is almost certainly true. Private schools don't grant employment contracts, for the most part, and Texas doesn't have a teacher's union that is allowed for public schools, so the private schools generally do not have them either.
If you are an "at will" employee, there is no were for you to appeal to. You can certainly send letters to the school President and there is no time restraint on that, so you can do it before the school year is out if you want, but that isn't a formal legal appeal process. There is no formal legal appeal process for an "at will" termination.
The only way that you've have any sort of recourse is if you have evidence that the basis for your termination is your race, religion, gender, age, disability or FMLA use. Then you'd need to be contacting the EEOC (or Department of Labor for FMLA use discrimination.)