How big a telescope do you need to see a 500-foot ship, 1 mile at sea?

It depends on what you are trying to see and resolve. 1 mile is not very far for such a ship, and with your naked eye you ought to be able to read its name! A good pair of binoculars should let you look at people's faces. And a 6-inch telescope ought to let you read a newspaper on-board.