I doubt we can predict future events with so little information: for example does Gates have medical insurance? Assume the patient has insurance: then the insurance company will have negotiated specific prices for a procedure, and that will probably preclude a billion dollar bill (although with fancy enough care, that could reach a billion dollars – we'd have to know the details of his policy to know who has to pay the charge). Point is, because of that contract, there won't be a special Bill Gates surcharge.
On the other hand, suppose the patient has no insurance. The hospital has a table of rates, known as the chargemaster, which is what they charge unless there is some agreement to a lesser amount (e.g. via Medicare or an insurance company contract). Here is one from Oakland CA. However, from a practical perspective the patient has to know that they are not just dealing with the hospital, they are dealing with physicians, respiratory therapists, lab technicians and so on, each of which will also charge something. In a non-emergency situation, a concerned citizen can inquire and decide whether they want to pay that amount; or whether they want to deal with a clinic which is evasive in answering questions and making promises about costs. Your question adds a minor complication, because his hospital actually subcontracts the actual tests to an outside lab, and the contract between the lab and the hospital would limit his liability for blood work.
There is a reasonable chance of counter-suing, for example on the grounds that the practice is unfair and deceptive (which is illegal in all states).
