The high observer says his clock reads 12:00, but he observes the lower clock as reading slower. The rotational position of the earth will be based on the lower clock reading, since that's the frame of reference the earth occupied.

No contradiction, just a matter of the relativity of simultaneity.

Of course, since you deny relativity, it's not something you'll ever understand.

Good answer, but I'd like to add something.

First, we need to define simultaneity. For a particular inertial frame, spacetime can be foliated into a series of hypersurfaces defined by constant time coordinate. Two events lying on the same "plane of simultaneity" are then deemed simultaneous. Of course, the time coordinate varies in between inertial frames, and so does the notion of simultaneous events.

This is the special relativistic description. In GR, the issue becomes more complicated. From MTW:

"In Newtonian theory or special relativity, one chooses hypersurfaces of constant time. But in dynamic regions of curved spacetime, no naturally preferred time coordinate exists. This situation forces one to make a totally arbitrary choice of hypersurfaces to use in visualizing the time-development of geometry, and to keep in mind how very arbitrary that choice was."

So, for two different observers in GR, we can't assign any notion of simultaneity - since we can't usually define a global frame of reference.

However, in local frames, GR reduces to SR, and so we obtain the usual description of simultaneity. Over short enough periods of time and intervals of space, or in weak fields, spacetime is roughly flat.

So, the answer to the original question - comparing the observations taken from two different points in curved spacetime yields no meaningful results. And thus there is no contradiction.

Note that this certainly does not mean we can't compare observations made at different points on Earth - its gravitational field is comparatively weak, and the effect of gravitational time dilation itself is not noticeable. However, the hypothetical posed is not even slightly local - and we can't compare those measurements.