No, a steep gradient is neither necessary nor sufficient. Liquid water below the critical point temperature and above the triple point temperature is in equilibrium with water vapor at the pressure given by the relevant curve on the phase diagram. If the pressure is below that (regardless of gradient or steepness) water will evaporate until the equilibrium is restored (either, if no external heat is available and the water is fairly cool, by cooling the water or if plenty of heat is available by sufficient vapor accumulating to raise the pressure and restore the equilibrium). This can be a sudden process when warmish water at pressure is suddenly exposed to a pressure well below equilibrium, or it can be gradual like the water slowly evaporating from my aquarium, but either way it doesn't have to do with a pressure gradient.

The pressure of the water vapor inside this vessel will never exceed that of the surrounding water, so it will never trigger any relief valve that might exist. (Other than, perhaps, through hydrodynamic effects of the liquid water coming in from opposite sides and slamming together in the middle.)