A medical doctor colleague suggested that it would be wise to have one's passport handy due to expected challenges in the US related to access to health care. For example, he indicated that someday it might be quicker to get bypass surgery done in India.
For people insured under Austrian law there are also bilateral contracts with other countries in place to enable tourists or delegated working personal to get regular health care (as the local regulation provides it). See also
If it's health care outside the EU and not during a holiday or as delegated working personal - eg seeking treatment options or expertise not available in Austria - you should get prior authorization from your health insurance. You need
proof of the indication
proof of the evidence for the (new) procedure
treatment history showing that other possible treatment options did not work or are not applicable