Planet–planet scattering is the leading mechanism to explain the broad eccentricity distribution of observed giant exoplanets. Here we study the orbital stability of primordial giant planet moons in this scenario. We use N-body simulations including realistic oblateness and evolving spin evolution for the giant planets. We find that the vast majority (~80%–90% across all our simulations) of orbital parameter space for moons is destabilized. There is a strong radial dependence, as moons past are systematically removed. Closer-in moons on Galilean-moon-like orbits (<0.04 R Hill) have a good (~20%–40%) chance of survival. Destabilized moons may undergo a collision with the star or a planet, be ejected from the system, be captured by another planet, be ejected but still orbiting its free-floating host planet, or survive on heliocentric orbits as "planets." The survival rate of moons increases with the host planet mass but is independent of the planet's final (post-scattering) orbits. Based on our simulations, we predict the existence of an abundant galactic population of free-floating (former) moons.