Can gravity cause a wave function to collapse? Assume the Copenhagen interpretation.
Suppose that a particle, for example an electron, has a wavefunction. If a heavy object, like the Earth, is close by, then that object interacts with the electron by gravity. Can this interaction cause the wavefunction of the electron to collapse? Has there been any experiment in which the wavefunction collapsed by gravity alone?
 A: I don't think it possible to observe. Wavefunction collapse is due to an irreversible interaction. Gravitational trajectories are a prime example of almost perfectly reversible, frictionless physics. Of course, gravity could deflect an electron wave to a detector, but we'd attribute the collapse to the irreversible detection process.
Edit:
Apparently, "gravitational decoherence" is the subject of theoretical study (thanks, @Connor Behan). Difficult to observe, though, for the reasons given above.
A: Here is a theoretical work on the wave function behavior of  neutrons in a classical gravitational field.

Spectra of neutron wave functions in Earth’s gravitational field

It is long and involved , expected plots for checking  a running experiment are given.
Here is an announcement at a conference with experimental checks .
Look at the paragraphs "Gravity Resonance Spectroscopy
with qBOUNCE"
So the answer to your  question:"Can gravity cause a wave function to collapse" is , yes, if by "collapse one means the wavefunction to interact with the gravitational field irreversibly.
This simplified model deals with the mathematics of coherence and incoherence : Coherence versus incoherence: Collapse and revival in a simple quantum model.

Unusual features found in the temporal behavior of this lossless model problem are "collapse," i.e., episodic nonexponential damping of both the atomic inversion and dipole moment, and two kinds of "revival" or partial recorrelation, in the dynamic evolution, during which the initial state is nearly recovered.

Italics mine.
The authors in the first link use this to correlate the bounce effect with data.
A: Yes. Measuring the system's sate by gravitational waves (gravitons) can cause wavefunction collapse (as we understand it). Measuring this effect is very difficult though.
