It's usually up to the hospital. If it's a tourist area, the hospital should be used to dealing with travel insurance and will be prepared to bill the insurance company. As long as you have the document with you, I cannot see too many problems in the US.

In some countries (e.g. Philippines) the hospital wont discharge a patient until the bill has been paid by someone. In the US they are likely to just check ID and make sure that there's a good chance of getting paid.