The work done by the force of gravity on a satellite moving around the Earth depends on the specific trajectory of the satellite. In general, the force of gravity does work on the satellite when there is a displacement along the direction of the force.
For a satellite in a circular orbit around the Earth, the gravitational force always acts perpendicular to the direction of motion. Since work is defined as the dot product of force and displacement, and the dot product of two perpendicular vectors is zero, the work done by gravity on a satellite in a circular orbit is zero. In this case, the gravitational force provides the necessary centripetal force to keep the satellite in its orbit, but it does not do any net work on the satellite.
If the satellite is in an elliptical orbit or any non-circular orbit, the force of gravity is not always perpendicular to the direction of motion. In such cases, the gravitational force does work on the satellite, and the amount of work depends on the specific trajectory of the satellite.
In summary, for a satellite in a circular orbit, the work done by the force of gravity is zero. For satellites in non-circular orbits, the work done by gravity can be nonzero, and its exact value depends on the specific trajectory of the satellite.