This paper addresses the controllability and global stability issues of a magnetically actuated satellite in the geomagnetic field. The variation of the geomagnetic field along the orbit, which is time varying in nature, makes the dynamics of the satellite time varying also. Sufficient conditions for controllability of such a time varying magnetic attitude control system are given. As a major contribution, it is proven that the three-axis controllability of the spacecraft actuated by the magnetic actuators is possible and it does not depend on the initial angular velocity of the spacecraft. Global controllability is a precursor to global stability. Therefore, exponential stability for an arbitrarily high initial angular velocity and an arbitrary initial orientation is proven next for a proportional-derivative control law using averaging theory. It is also proven that even an iso-inertial satellite can be stabilized using the time invariant feedback control, which was hitherto not possible, even using time variant conventional control. Simulation results are presented under different initial orientations and angular velocities of the satellite in the presence of favorable and unfavorable gravity gradient torques to validate the proposed control method.