Can I wire a refrigerator and dishwasher on the same circuit?
Maybe.
Branch Circuits Required

National Electrical Code 2014
Article 210 - Branch Circuits
I. General Provisions
210.11 Branch Circuits Required.
(C) Dwelling Units.
(1) Small-Appliance Branch Circuits. In addition to the number of branch circuits required by other parts of this section, two or more 20-ampere small-appliance branch circuits shall be provided for all receptacle outlets specified by 210.52(B).
III. Required Outlets
210.52 Dwelling Unit Receptacle Outlets.
(B) Small Appliances.
(1) Receptacle Outlets Served. In the kitchen, pantry, breakfast room, dining room, or similar area of a dwelling unit, the two or more 20-ampere small-appliance branch circuits required by 210.11(C)(1) shall serve all wall and floor receptacle outlets covered by 210.52(A), all countertop outlets covered by 210.52(C), and receptacle outlets for refrigeration equipment.
Exception No. 2: The receptacle outlet for refrigeration equipment shall be permitted to be supplied from an individual branch circuit rated 15 amperes or greater.

210.52(B)(1) Exception No. 2 specifically says that you can put the refrigerator on a dedicated circuit (that can be 15 amperes instead of 20), but does not require it.  The rest of 210.52 says that you have to have at least two 20 ampere branch circuits to supply all the required receptacles in the kitchen, and that those circuits can only serve receptacles in that same kitchen (with some minor exceptions).
If you can meet the provisions in 210.52, and still end up under the load calculations in other parts of the code.  AND the dishwasher is cord-and-plug connected, then it is possible to have the fridge and dishwasher on the same 20 ampere small appliance branch circuit. Whether or not this is recommended is subjective, and can be debated in another place.
Load Calculations

National Electrical Code 2014
Article 210 - Branch Circuits
II. Branch-Circuit Ratings
210.23 Permissible Loads.
(A) 15- and 20-Ampere Branch Circuits.
(1) Cord-and-Plug-Connected Equipment Not Fastened in Place. The rating of any one cord-and-plug-connected utilization equipment not fastened in place shall not exceed 80 percent of the branch-circuit ampere rating.
(2) Utilization Equipment Fastened in Place. The total rating of utilization equipment fastened in place, other than luminaires, shall not exceed 50 percent of the branch circuit ampere rating where lighting units, cord-and-plug connected utilization equipment not fastened in place, or both, are also supplied.

This means that the refrigerator must be rated less than 80% of the branch circuit rating,  for it to be connected to a 20 ampere small appliance branch circuit. This would mean the fridge must be rated less than 16 amperes.
20 amperes * 0.80 = 16 amperes
It also means that the dishwasher (assuming it's not a portable unit), must be rated less than 10 amperes.
20 amperes * 0.50 = 10 amperes
If it's a portable dishwasher, it can be rated up to 16 amperes just like the refrigerator.

tl;dr
So if you have:

A cord-and-plug attached dishwasher fastened in place that is rated 10 amperes or less.
Or a cord-and-plug attached dishwasher not fastened in place that is rated 16 amperes or less.
And a cord-and-plug attached refrigerator not fastened in place that is rated 16 amperes or less.
And you meet the provisions in 210.52(B)

Then it is possible (though maybe not advisable) to have a dishwasher, and a refrigerator on the same small appliance branch circuit.