With enough epicycles, the motions of the planets can be accounted for in a geocentric model, but do the laws of gravity make such motions impossible? Did Newton's laws not only provide a simpler heliocentric alternative, but also completely rule out the possibility of geocentric motion?
They can these days.
Try explaining the behavior of interplanetary spacecraft in terms of a epicycle model. Just be aware that we can change the cycles and epicycles at will by running the rocket on the probe (or deploying a solar sail).
By contrast a model based on Newton's laws of mechanics and gravitation handles these situations smoothly.
In principle in Newtonian mechanics the rest frame can be any of the bodies in a gravitational complex. The geocentric system is one possible rest frame and a one to one mathematical transformation exists going from a heliocentric to a geocentric system. It is when one introduces the concept gravitation, a theory that explains the orbits, that the heliocentric system simplifies and explains the epicycle motions.
Newtons laws assign forces that describe the motions mathematically. The geocentric model without masses has no explanation for the complicated paths of the celestial bodies except the geometric rules. A theory is by far better than a phenomenological fit to observations. Further more the expressions if one sticks to a geocentric Newtonian rest frame get complicated unnecessarily, as dmckee points out for satellites etc.