We know that sometimes being a student and trying to keep a descent GPA can be really hard.
In the classical problem it's important that the ball was dropped from rest, otherwise the time taken to hit the ground will vary.
But the quantum solution doesn't involve the initial conditions at all. My first thought was that I need the time-dependent Schrodinger equation instead of the time-independent one, but that doesn't lead anywhere - it just means the ball oscillates between solutions.
It would occupy some standing wave about the Earth and not evolve in time.
The reason we never see macroscopic objects in such states is because they are unstable in the same sense as Schrodinger's cat.
Your proposed stationary states would delocalize the ball over the entire Earth, over a scale of thousands of miles.
But if you just at the ball, you can measure where it is, collapsing this superposition.
constructs a solution that obeys Newtonian gravity.
You're of course correct that if the ball were in a single stationary state, then it wouldn't do anything remotely like falling.
The point is that, quantum mechanically, the initial conditions cannot be a position $x_0$ and momentum $\xi_0$ at a fixed time.
In classical (statistical) mechanics, the initial condition is a probability distribution in the phase space (in the case we are considering, it is a delta distribution centered in the initial condition $(x_0,\xi_0)$).