If you're driving anywhere outside the U.S., your American auto insurance coverage is typically not in effect. Sometimes it is in effect in parts of Canada and within a few miles of the southern U.S. border.
If you are renting a car in Europe or elsewhere, you should purchase car insurance from the foreign rental car company at levels similar to your auto insurance coverage in the U.S.or at levels required in that area. Make sure that the car insurance provided by the rental car company complies with auto insurance regulations in all the countries where you plan to drive.
If you are not renting a car, but rather borrowing a car. You should contact the carrier that covers the borrowed car.
Comment Update: You are correct, using your own car is unrelated to renting a car.