If you are an at will employee, your employer can terminate you for any reason or even for no reason. It certainly can terminate you if it believes you falsified documents, even if you didn't and even if you could absolutely prove you are innocent of those charges. As long as the termination was not motivated by the fact that you are in a protected class of people, or because you engaged in protected conduct, you can be fired even if the decision is irrational, unreasonable or based on wholly false facts.
As to suing the union. That will likely be a non-starter. It is not the entity that fired you. A claim of ineffective assistance by the union will not work either.
Good luck finding a new job.
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In addition to what Neil said, below is some information on at will employment. Wrongful termination cases are tough to win. You might have some breach of the collective bargaining agreement, that could support a breach of contract. You might be able to sue the union for breach of duty of fair representation, but that's not an easy case to win, especially in San Diego.
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