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In the classical Hall effect a magnetic field deflects a charge-carrier moving through a conductor to the edge of the conductor, causing a transverse potential across the conductor.

Does gravity not do the same thing? In a horizontal wire, free electrons should be deflected to the bottom edge of the wire, causing a potential vertically across the wire.

Since gravity does not act on moving charge carriers only, no current should be required to see this effect. A horizontal metal plate should have a potential across it from free electrons drifting down until force of electrostatic attraction pulling them back into the conductor cancels the gravitational attraction.

Is this simply wrong? Or too small to measure?

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Gravity does indeed have the effect you describe, and actually it's very easy to calculate how big the effect is.

Suppose you have a conductor with some height $h$. Then if an electron moves from the top of the conductor to the bottom it gains a gravitational potential energy $mgh$.

But the excess of electrons at the bottom of the conductor produces a potential difference $V$ between the top and bottom of the conductor, and to move against the potential difference costs an energy $eV$.

The system is in equilibrium when the two energies are the same, that is:

$$ mgh = eV $$

and a quick rearrangement for the voltage developed due to gravity gives:

$$ \frac{V}{h} = g\frac{m}{e} $$

The electron mass to charge ration is about $5.7 \times 10^{-12}$, so the potential gradient due to the gravity at Earth's surface is about 56 picovolts per metre.

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Electrons are quantum mechanical entities. Newtonian gravity is classical. One can apply a gravitational potential in the quantum mechanical equation describing an electron, and if it is a free electron, there will be the gravitational attraction that all macroscopic masses feel, but it will be very small due to the weakness of the gravitational field. With respect to the electromagnetic attraction the difference is enormous, order of $10^{-37}$ smaller. Some effects of gravitation can be seen in collective electron beams, but it is from changes in the total boundary conditions.

Electrons in solid matter (or liquid or gas) are bound in energy levels quantum mechanically, and any effect of an external potential will affect the energy levels, but again, the gravitational potential is very small to make a difference. What you consider as free electrons are in energy bands, bound to the lattice, and the bands will be different infinitesimally if on Earth or on the Moon, but not to make any measurable difference.

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