the Michelson-Morley experiment proves Earth isn't moving
The physical fact established by the Michelson-Morley experiment is that the speed of light at the surface of the earth is isotropic. This is the famous "null" result of the experiment.
This fact can have two different interpretations depending on which speed of light we think was measured by the experiment:
- the speed of light in space
- the speed of light inside the earth's atmosphere
If we assume that the experiment measured the speed of light in space, then the "null" result of the experiment implies that:
- the earth is not moving through space, as only observes at rest with
respect to an optical medium should measure an isotropic speed of
- provided that the earth is in fact moving through space, the "null" result of the experiment then implies that the speed of light is independent of the motion of the observer, as apparently even observers in motion with respect to the optical medium measure an isotropic speed of light
So, yes, the MM experiment does imply that the earth is not moving through space.
On the other hand, if we assume that the experiment measured the speed of light inside the atmosphere, then the "null" result of the MM experiment shows that the speed if light inside the atmosphere remains isotropic while the earth is moving through space. Therefore, this speed is relative, in the Galilean sense.
This is actually Galilean Relativity 101: you cannot detect the motion of a Galilean ship from inside the ship; for that, you need to go outside. Same applies for the earth and its atmosphere.
If you want to detect the motion of the earth through space, you need to measure the speed of light outside the atmosphere.
One way to do this, as you correctly alluded to, is to repeat the MM experiment in orbit. This has not been done yet.
The alternative is to take a look at the satellite missions for mapping the Cosmic Microwave Background (CMB) - and in particular at the "dipole anisotropy":
The "dipole anisotropy" is the closest that we have to replicating the MM experiment is space.