Actually I think the formula is derived empirically, and the value is in knots not mph. From an old typewritten NASA technical memorandum (NASA-TM-85652):
The contact pressure between tire tread and pavement establishes the escape velocity of bulk water drainage from beneath the tire footprint. High pressure tires can expel surface water more readily from the footprint than low pressure tires. When the aircraft ground speed equals or exceeds the escape velocity of water drainage from the footprint, choked water flow occurs. The tire has now reached a state of total dynamic hydroplaning. Test results .... indicate that the critical aircraft groundspeeds required for this total hydroplaning condition to occur on a flooded (runway water depth is greater than tire tread groove depth) pavements with an unbraked tire are approximately:
Spin-down (rotating tire) speed, knots = 9 sqrt Infl. pressure, psi
Spin-up ( nonrotating tire) speed, knots = 7.7 sqrt Infl. pressure, psi
......... It is important that pilots be aware that the lower hydroplaning spin-up speed, rather than the high hydroplaning spin-down speed, represents the actual tire situation for touchdown on flooded runways.