It’s impossible to even reason about what something entirely unaffected by gravity might or might not do because everything, including gravity, is relative. Something not affected by Earth’s gravity would probably get carried away by the wind, but something not affected by the Sun’s gravity would float away from the sun (relative to the Earth) and appear to have some sort of mysterious acceleration from our perspective. If the object isn’t affected by the Milky Way’s gravity, even more shenanigans.