How can I contest wrongful termination in new york state

The answer depends upon your employment status. If you are not under a contract with your employer or in a union, you can be fired for any reason or indeed for no reason (which would include a mistaken or incorrect reason). NY is an "at-will" state - that means you can quit without giving notice and you can be fired without notice. Of course, you cannot be fired for a constitutionally-protected reason (age , gender, national origin and in NYC sexual orientation for example) Generally, unless you are in a union or under contract, in NY State you are an "at-will" employee and can be fired for no reason and without notice. Sometimes, you may have a claim if you have been fired in a manner that violates a written policy manual, though most of those contain a statement that tells you that you are an "at-will" employee regardless of what else the manual says.