I have a ship centered at the screen with a radius of 10 from the center of sphere located at (0, 0, 0) ... I am using the standard on-screen Joystick provided in the Standard Assets (Mobile) package of Unity. Can someone help me figure out how to allow the joystick to move the ship around the sphere by a specified radius? I currently have the joystick controlling the angle the ship is rotated at but I can't seem to get it to move the ship in a correct orbit.

You will have to use way points to know where exactly it should travel or create an algorithm to move along a desired motion.

Or actually it can be done easier thanks to Unity. http://forum.unity3d.com/threads/20536-Moving-camera-on-a-circle-path