Your tenant could file a lawsuit aganist you if in fact he was injured due to your negligence. It would make no difference if he was moving out on the same day that he was injured. I am not sure what the applicable landlord/tenant law is in your state. You would need to consult with a local attorney to find out what your rights are as a landlord.
I think that you need to put your insurance carrier on notice that the tenant is claiming that he was injured on your property.