This is a very unfortunate situation and it appears quite clear that you were released for reasons that are not justified. That said, I must tell you that employment in the state of NY is "at will" absent an express agreement to the contrary. At will employment can be terminated for any reason not amounting to discrimination on the basis of a legally protected trait (race, religion, gender, etc.) or retaliation for engaging in certain forms of legally protected conduct (filing a wage claim, taking FMLA leave, etc.). It doesn't matter whether the basis for termination is fair, reasonable or even true.
Thus, you can be terminated because of a dispute in which you were not the instigator, even if you did absolutely nothing wrong. Your employer owes no legal duty to investigate or to maintain your employment. If, for any reason other than a protected trait or activity (as defined above) they no longer want to continue the employment relationship, they do not have to continue it. This freedom works both ways and is the reason you can quit without notice to your employer if you are ever so inclined.
I'm afraid here, given the lack of protection afforded by the law, your only remedy is to attempt to reason with your employer that their decision was not justified. This would typically mean writing a letter or email to the HR department and explaining your side of the story and why you believe that there was no legitimate reason to end your employment. They will not be legally required to reconsider your termination, but since it costs employers time and money to hire and train replacements, most employers have a self interest in only terminating employees when there is a good reason for it. Thus, most employers will consider a letter that you write to them asking for a second chance if you can give them a good reason.
If I can clarify anything at all for you, please do not hesitate to ask. It is my pleasure to assist you further if necessary....