Proof that the Earth rotates?

The Foucault pendulum is a great experiment which does demonstrate that the Earth is rotating, but it was only introduced in 1851. The Earth had been known to rotate for several centuries before that, probably stimulated by Copernicus and Galileo pushing the heliocentric model of the solar system during the 16th century.

A couple of decades before Faucalt's pendulum, the Coriolis effect was discovered. This effects (among other similarly large systems) hurricanes, causing them to rotate clockwise/anti clockwise depending on whether they're in the southern/northern hemisphere. It is an apparent force that appears in any rotating frame of reference (like a spinning planet). This again won't have helped early 'spinning-Earth' believers.

Early evidence that the Earth rotates was almost certainly the observation of the sun, planets and stars moving across the sky and then, with the help of telescopes, of the other planets also rotating. Of course this requires you to trust that the Earth is not the centre of the universe and so doesn't "prove" the Earth is spinning in the same way as observation of Foucault's pendulum or the Coriolis effect.

The idea of a "proof" in physics is a difficult one. A theory such as the 'spinning-Earth' will at first simply be presented to explain anomalous observations that current theories can't (such as why do the other planets and the sun move independently of the background stars if everything rotates together around the Earth?). It then gets tested by experiments inspired or predicted by the theory (if the Earth is spinning, what should we expect to observe?). If everything holds up, it's accepted as fact. Eventually, some bright spark realises he/she can demonstrate that the Earth is rotating with a clever little experiment (Foucault's pendulum/Coriolis effect/launching a rocket into space) and it is added to the mountain of evidence already accumulated on the subject.

Foucault pendulum. I don't know how the ancients did it, but it is surely pure classical mechanics.

The animation describes the motion of a Foucault Pendulum at a latitude of 30°N.

I think the Foucault pendulum is the best answer, but for the sake of variety I'll add another very interesting one: the equatorial bulge affecting the figure of the Earth. This is the "pancaking" of the planet due to its rotation. You can measure the geometry of the Earth without leaving its surface, and find that it is bulging in accord with your expectations if that bulge were caused by rotation at the same rate as the one we observe relative to distant stars. As always, there's no such thing as "proof" in physics, but this is strong supporting evidence.