In the first instance and in the majority of medical diversions the airline pays. A passenger becoming ill does not necessarily result in a diversion. The Captain would make the final decision on the course of action to be taken. This would occur after seeking whatever guidance and advice was available in the circumstances, including professional guidance and advice that may be available after radio communication.
It is not unusual for passengers taken ill on cruise ships to be charged for the costs involved in treating them or removing them to another facility, and although it seldom happens on airliners, there is nothing in practice preventing airlines from seeking recovery of their costs in such circumstances. All travellers are recommended to carry insurance to protect them against the costs incurred in falling ill abroard or in transit, and such insurance should be able to meet the expenses resulting from a decision to divert. In the future I suspect this is likely to happen more and more.
In the case of a diversion due to a passenger causing a disturbance, that would be matter for compensation in the criminal courts, or recovery through the civil courts depending on the circumstances and outcome. Certainly it is covered in the contract of carriage that a passenger would be deemed liable for such costs.