The rules are essentially same whether FAA or ICAO.
FAA Order 7110.65V Air Traffic Control
2−1−4. OPERATIONAL PRIORITY
Provide air traffic control service to aircraft on a “first come, first served” basis as circumstances permit, except the following:
a. An aircraft in distress has the right of way over all other air traffic. 14 CFR Section 91.113(c).
b. Provide priority to civilian air ambulance flights (call sign “MEDEVAC”). Use of the MEDEVAC call sign indicates that operational priority is requested. When verbally requested, provide priority to AIR EVAC, HOSP, and scheduled air carrier/air taxi flights. Assist the pilots of MEDEVAC, AIR EVAC, and HOSP aircraft to avoid areas of significant weather and turbulent conditions. When requested by a pilot, provide notifications to expedite ground handling of patients, vital organs, or urgently needed medical materials.
NOTE−
It is recognized that heavy traffic flow may affect the controller’s ability to provide priority handling. However, without compromising safety, good judgment must be used in each situation to facilitate the most expeditious movement of a MEDEVAC aircraft.
c. Provide maximum assistance to SAR aircraft performing a SAR mission. ...
You seem to view things in black & white and think that just because you used the word "MAYDAY" on a passenger medical emergency, that gives you the right to ignore ATC instructions, bust clearances, deviate your flightpath, and do whatever you want.
Sorry, but doing so would be exceeding your authority as PIC, putting others at risk, and might even delay care for the passenger in distress.
Not all emergencies are equal. Consider:
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Aircraft A calls ATC with "MAYDAY, passenger medical emergency, request immediate diversion to Cairo"
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Aircraft B also calls ATC with "MAYDAY, fire on board, losing power on both engines, request immediate diversion to Cairo"
By law, which aircraft has priority? Which pilot may deviate from ATC clearances?
PIC authority to deviate under emergencies are limited by statutes only "to the extent required to meet that emergency".
The pilot of
Aircraft B would be well within his rights immediately "turn inbound Cairo VOR" and do whatever it takes to get his aircraft safely on the ground, ATC clearances be damned.
But the pilot of
Aircraft A would NOT have the same right to also bust clearances, turn into the Cairo VOR, and do "whatever" to land at Cairo without permission. Such actions would be out of the norm for an in flight medical emergency, and might be considered well in excess of what is required given the nature of the emergency.