Even if you are undocumented, you can file a lawsuit for personal injuries in New York
In a 2005 decision out of Nassau County Supreme Court Judge Ira B. Warshawsky allowed not only for illegal aliens to receive compensation for medical benefits and lost wages as well as pain and suffering, but for proven future lost earnings as well. Echeverria v. Estate of Lindner 7 Misc.3d 1019(A), 801 N.Y.S.2d 233 N.Y.Sup.,2005. Therefore, undocumented aliens need not fear bringing a law suit to recover money damages for personal injuries.
You should not fear threats of deportation
It is not legal to threaten prosecution of a criminal matter to gain an advantage in a civil matter. Additionally, the fact that you are undocumented is irrelevant to your claim.