It's been a while since fluid dynamics but gravity matters for the force balance. In the force balance, you're going to have gravity, frictional forces from the surface/shear stress of the water, and a pressure applied to the fluid from some direction to complete the balance.
I didn't see anything in the question indicating the direction of flow but the easiest is probably going to be to assume that the flow is going downward or upward. Gravity is acting in a downward direction so either a pressure or a frictional force has to balance with the gravitational force acting in the vertical direction or there's going to be acceleration in that direction. Since the flow is steady, it's not accelerating. If you make the flow upward then you need a pressure acting in the upward direction which then counters the force of the friction and gravity together.
If the flow is vertical you'll still have pressures along the side so as to ensure flow is moving in a direction but you can just say there's a pressure acting there and the two sides balance to zero making it meaningless in the force balance.
If the flow is horizontal across the plate then you'll have a frictional force in that direction that you will have to account for by some pressure in addition to the vertical forces that remain in place.
I'd have to refer to a textbook to break down the shear stresses and velocity profiles as well as the equation you've used since I don't remember all of the terms offhand but everything should fit within the framework of that force balance.
Realistically whether the flow is horizontal or vertical it isn't harder it's just which terms you're claiming are at a balance with the others.