This story filled me with pride for the Navy and the abilities of my beloved Nimitz-class carriers (I served on the Nimitz). A 14-year old girl suffers a ruptured appendix while at sea and the Reagan responds to the call. Read about it here.
I do know that even in the military the commanding officers have budgets. However this mission was approved by Commander, Third Fleet, so I am thinking the public relations the Navy gleaned from it will be payment in full. Plus, can you imagine the negative press if the Navy sent a bill?
If the fleet is already at sea conducting military exercises, I wouldn't think diverting in response to a medical emergency would be a big deal.
It is a big deal when it causes a disruption in the exercise. That could impact thousands of people involved in the exercise and cost $$$$$ because they have to put things on hold. In this case I haven't heard anything about the billing so assume the Navy took their payment in the form of good press about the story.