His 'sound' (although slightly influenced by the hardware) comes from his fingers. He doesn't fingerpick the traditional way (traditional: allowing the string to vibrate cleanly with no string and fret interference noise)
He uses a more aggressive motion, pushing strings toward the body with his right hand fingers which causes the string to purposely rattle against the fretboard. This rattling effect, when done cleanly and with proper eq, give the Geddy growl. His fingers are in very good shape and to play with this technique for any length of time, they have to be. I guarantee a Rick, a Jazz, or a Steinberger won't give you the "Geddy" sound as much as the above technique will.
Good luck.