This is quite an interesting question actually. How can we usually judge the rotation of some abstract field? By moving its power lines. However, the problem is that no one has ever seen the lines of force of the earth’s gravitational field (and indeed any gravitational field), no instruments have registered, and it is difficult to even guess the direction. For this reason, it was even decided to assume that the gravitational field as such does not exist in nature. And there is a curvature of space around massive bodies.
So, to paraphrase the question, does the Earth’s gravitational field rotate (the curvature of space)? Fortunately for us, the shape of the Earth is very different from the ideal ball, so its gravity is also not uniform at all. These “dips” and “peaks” are felt even by artificial satellites of the Earth. And it is quite clear that these inhomogeneities move after the daily rotation of the planet. So, even without seeing the lines of force and spatial “pits”, we can state with full confidence that the gravitational field of the Earth rotates together with the planet itself.
