Am I responsible for damages to another vehicle in a no-fault state?

If I nicked a car and the driver won't claim it, is it my responsibility to pay? I live in a no-fault state (Michigan).

Although you're right and Michiganis ano-fault state, this refers only refers to bodily injury claims. Things like physical protection to your vehicle aren't covered under this definition. Because you were at-fault for the accident, if she wants the damages repaired, it is your responsibility to pay for the damages. However, that is only if she chooses to file a claim.