The moon probably doesn't cause earthquakes, but Earth might cause moonquakes! For a full explanation, we asked Geza Gyuk, Director of Astronomy at the Adler Planetarium and a research scientist at the University of Chicago.
Here's what he said:
The short answer is that it is possible, but no convincing evidence has ever been shown to connect the moon's influence to seismic activity.
While the moon does exert a gravitational force on the Earth, the strength of that force is very small, less than a millionth as large as the force of gravity from the Earth itself. Add to this the fact that the Earth is constantly shivering and shuddering with earthquakes due to the much larger stresses of plate tectonics and the movements of molten rock deep within. Perhaps it isn't any wonder that earthquakes due to the moon have never been convincingly demonstrated.
Quakes due to another celestial body are not entirely impossible though. We've even measured some moonquakes due to the Earth.
The busy Apollo astronauts left behind more than their boot-prints when they visited the moon back in the 1970's. They left some sensitive seismometers.
The moon is a practically dead object, with no volcanism, plate tectonics or even weather. Almost nothing moves on the moon, so it is ideal for detecting tiny moonquakes due to the very small effect of the gravity of the Earth. It helps that the Earth is much more massive than the Moon and so the gravitational force is much greater.