In the case of a satellite in a circular orbit, speed2 / r = acceleration from gravity at that distance (r) from the center of earth, so it follows a circular path. In this case the acceleration only changes the direction of velocity, without changing it's path or speed. The satellite is always accelerating towards the earth, but it's velocity perpendicular to that acceleration causes it to remain in a circular path.
If the orbit is elliptical, then most of the time gravity is not perpendicular to velocity, and speed will change, fastest when closest to earth, slowest when farthest away from earth. In a two body system, (earth and satellite as only two objects in the universe), the center of mass of the two body system doesn't accelerate.
Which fundamental source is centripetal doesn't matter. In the case of a satellite in a circular orbit, it's gravity, but the force could be magnetic (a moving charged object curves perpendicular to a magnetic field), or mechanical force like a test tube in a centrifuge.