It wouldn't be shot "down."
The projectile velocity vector would be the sum of the force firing it toward Earth combined with the angular momentum of the satellite that shot it. Just to maintain orbit something needs to travel about 8k per second so that force vector is added tangentially to the projectile's firing vector… Eg it will trace an ellipse not go "down"
Depending on "how hard" you shot it you could end up with the projectile following that ellipse and actually hitting whatever it was shot from.
This all ignores the fact that you'd have to fire it with a ridiculous amount of force, ignoring orbital mechanics, to make it do comparable damage to a conventional warhead.