If the center were identified with the center of the Earth, then the theory could be taken to eschew absolute quantities: it would simply hold that the natural motions of any body depend on its position relative to another, namely the Earth. But Aristotle is explicit that the center of the universe is not identical with, but merely coincident with the center of the Earth (e.g., On the Heavens II.14): since the Earth itself is heavy, if it were not at the center it would move there! So the center is not identified with any body, and so perhaps direction-to-center is an absolute quantity in the theory, not understood fundamentally as direction to some body (merely contingently as such if some body happens to occupy the center). But this conclusion is not clear either. In On the Heavens II.13, admittedly in response to a different issue, Aristotle suggests that the center itself is ‘determined’ by the outer spherical shell of the universe (the aetherial region of the fixed stars). If this is what he intends, then the natural law prescribes motion relative to another body after all — namely up or down with respect to the mathematical center of the stars.
In fact, I think the position they note in the last two sentences is Aristotle's account; in an astronomy based on the geometry of circles, the center is just defined by which circle you are using, so the only question is whether there is a privileged circle, a circle that encompasses all the rest, whose center is the center of everything. But the center itself is always going to have to be the center defined relative to at least some significant circle.