Does shutting your finger in the car door constitute a car accident?

My husband stepped out of his truck and in the process shut his index finger in the door. It was bad enough that he had to go to the ER to get it checked out. Now the hospital is billing us...but our medical insurance won't pay because they are insisting this is a "car accident". Our car insurance provider insists this is a medical issue and health insurance should pay. In the meantime we are getting dunning notices from the hospital. Who's right and who is wrong? Which insurance is liable?