  The interplay between quantum theory and general relativity remains one of
the main challenges of modern physics. A renewed interest in the low-energy
limit is driven by the prospect of new experiments that could probe this
interface. Here we develop a covariant framework for expressing post-Newtonian
corrections to Schr\"odinger's equation on arbitrary gravitational backgrounds
based on a $1/c^2$ expansion of Lorentzian geometry, where $c$ is the speed of
light. Our framework provides a generic coupling prescription of quantum
systems to gravity that is valid in the intermediate regime between Newtonian
gravity and General Relativity, and that retains the focus on geometry. At each
order in $1/c^2$ this produces a nonrelativistic geometry to which quantum
systems at that order couple. By considering the gauge symmetries of both the
nonrelativistic geometries and the $1/c^2$ expansion of the complex
Klein--Gordon field, we devise a prescription that allows us to derive the
Schr\"odinger equation and its post-Newtonian corrections on a gravitational
background order-by-order in $1/c^2$. We also demonstrate that these results
can be obtained from a $1/c^2$ expansion of the complex Klein--Gordon
Lagrangian. We illustrate our methods by performing the $1/c^2$ expansion of
the Kerr metric up to $\mathcal{O}(c^{-2})$, which leads to a special case of
the Hartle--Thorne metric. The associated Schr\"odinger equation captures novel
and potentially measurable effects.
