No, you really can't.
Quote by: tinybear
Section 10 of the WCL establishes that employers have to have Workers' Comp insurance and must cover the medicals and out-of-work expenses of employees who are injured or die while in the course of their employment.
Then, Section 11 says this:
NY Workers' Compensation Law, s.11
The liability of an employer prescribed by the last preceding section shall be exclusive and in place of any other liability whatsoever, to such employee, his or her personal representatives, spouse, parents, dependents, distributees, or any person otherwise entitled to recover damages, contribution or indemnity, at common law or otherwise, on account of such injury or death or liability arising therefrom,(emphasis added)
If you're injured or die on the job, your exclusive remedy is to file a claim for Workers' Compensation, not a lawsuit.