Did he believe stars were attached to the sphere of fixed stars, or did he believe they were suns or did he believe stars weren't attached to anything but just points of light?
I'm not sure Galileo had any solid belief regarding the nature of astronomical objects.
Copernicus posited the "immobile sphere of fixed stars" in his On the Revolutions of Celestial Spheres in 1543, and in 1651, we have the publishing of The New Philosophy of Our Sublunary World, said to have been posthumously compiled from the notes of William Gilbert, which included an argument that the stars were not fixed to a sphere but rather at varying distances (all extremely remote). Gilbert died in 1603, so assuming the attribution of New Philosophy is correct, there was already doubt as to the "crystal spheres" hypothesis while Galileo was still teaching at the University of Padua, many years prior to his major astronomical discoveries.
Galileo was certainly a follower of Copernicus in the sense that he accepted that the Sun was the center of the solar system, and by observing the phases of Venus, he proved that Venus must orbit the sun and not the Earth. But I don't know how much he bought into the celestial spheres theory, and the answer may well have changed during his lifetime.
The early 1600s was a time of rapid discovery in the field of astronomy, and Galileo was an observational scientist deeply involved in the advancements of the time, so I doubt he was particularly attached to any one explanation for the things he was seeing. It seems to me that Galileo was driven primarily by what he saw, and willing to entertain any idea for long enough to decide what observation might support or disprove it, and make such an observation.