If you think the reason was just an excuse to cover up discrimination, you should consult an employment lawyer. Otherwise, unless you have a union, you can only try to collect unemployment.
Unless you had a contract, you were an at-will employee and therefore could be fired for any reason or no reason so long as it was not an unconstitutional reason such as based on race, gender, religion, etc.
The above answer, and any follow up comments or emails is for informational purposes only and not meant as legal advice.
Unless you feel the termination was motivated by some form of discrimination or retaliation then there is not much my firm could do to assist you. I am sorry.
This answer does not constitute legal advice and you should contact an attorney to confirm or research further any statements made in this answer. Any statements of fact or law I have made in this answer pertain solely to New York State and should not be relied upon in any way in any other jurisdiction.
I agree with my colleagues - even IF the person lied, unless there is some discriminatory intent, or unless you are a Union employee or an employee with a contract, they can fire you for any reason. There is nothing that most attorneys can do to help.
I also don't think it is worth your time suing the employee that likely lied about your tardiness, because that employee probably doesn't have any money to pay a judgment, and you'd be spending more than you would get on lawyer fees and court filings.
Sorry I couldn't give you better news.
This does not constitute legal advice or the engagement of my services as an attorney.