If you are in a foreign country (foreign to you), your insurance policy doesn't (usually) apply, unless it's Canada. If you have insurance in THAT country, maybe the accident would be covered.
If you're in a country with socialized medicine, you need to be aware that if you're not a citizen/legal resident, with the proper paperwork showing you HAVE rights to their health care system, they not only won't pay to treat you, but they CAN turn you away, at the hospital, if they want.
No idea "how insurance company behaves", as you haven't specified that there IS one that covers you.
If you rent a car and have an accident, if you didn't buy the insurance from the car rental place, they will charge all the damages to the credit card you used to rent the car.
the insurance you paid for when y ou got there handles it all/went to italy/rented car from avis in usa/took care of it here/including accident and health/be prepared
Providing you are insured it's the insurers problem not yours, how ever, it will cost you quite a bit more money when you re-new your insurance and it doesn't work to change companies either.