Can I split the direct wire to dishwasher to handle a garbage disposal as well?
I only have 2005 NEC handy, but there's nothing there against using same circuit for both dishwasher and disposal. 

210.23(A) Branch circuits, rated 20 or 15 amperes may feed lighting units, utilization equipment or combination of both. The rating of of any one cord and plug connected utilization equipment shall not exceed 80% of the branch circuit ampere rating.

Your local code could be different, of course. 
Also note, that your garbage disposal needs a disconnect switch