Let's see... you suppose that water was ejected from the earth at something like seven miles per second (which would be necessary to get it "into the sky.") The friction of such velocities would produce a huge quantity of heat.
BTW, Adiabatic cooling is the process of reducing heat through a change in air pressure caused by volume expansion. Water, being nearly incompressible, will not exhibit such cooling, unless heated to a vapor. Water does have an extraordinarily high specific heat, which would help it absorb the vast amounts of thermal energy produced by the velocities you're imagining.
At those energies, even ice would quickly be vaporized, and much of the thermal energy would be contained in the water vapor initially.
Thermodynamics (yes Stipe, that thing that baffles and annoys you) requires that thermal energy move from hot objects to cooler ones. So the superheated water vapor would then by conduction, radiation, and convection transfer thermal energy to the atmosphere, steaming the biosphere.
Assuming the quantities and velocities you have imagined for the ejected water. Merely ejecting water at high pressures does not vaporize it, of course. If that were true, water cutters would not be possible. But accelerating water through the atmosphere at close to escape velocity would certainly do that, producing a huge amount of heat.
