In response to the Qur'an's claim that Allah fixed the earth, Muslims have usually concluded that the Qur'an is referring to the earthly crust, the surface resting nicely on the mantle.
However, the Qur'an does make a clear reference to this:
Allah grasps the heavens and the earth lest they move from their places’ (Sura 35: 41)
Hence the fixing and resting of the earth is not the earth's crust or surface but the planet itself which in correlation with the heavens are being held in fixed place.
Then we might discuss whether the passage reveals fixation within a fixed orbit or the earth's attachment to the sun.
There are however, two difficulties here, the passage does not clarify this and such amounts to speculation. Furthermore, the Qur'an is clear not only from its metaphorical language but also its scientific description that the sun orbits around the earth.
This presumes that the earthly body is 'cosmological central' and fixed, much like space itself (from the Qur'anic description).