As you point out, the spell itself doesn't specify whether the water is moving. If it was moving, it would likely be specifically called out in the spell text.
For example, Tsunami, a similar water creation spell, states
A creature caught in the wall can move by swimming. Because of the force of the wave, though, the creature must make a successful Strength (Athletics) check against your spell save DC in order to move at all. If it fails the check, it can’t move.
It also explicitly pushes creatures around. Neither of these abilities are enumerated in the text for Wall of Water.
As a side note, you could potentially get the water moving by casting Shape Water on the resulting wall of water. It might be a bit controversial, but I'd rule that it would work...