Connecting op amp circuit output to ADC makes signal asymetric
If you look at the equivalent circuit of an LM324 you will see why it cannot pull down to Ground under load. Output transistor Q13 is connected in Emitter Follower configuration, so when sinking current the Emitter will be ~0.6V above the Base (which at best can almost reach ground potential through Q12). There is a current sink in parallel with Q13 which can pull down further, but only at very low current (&lt;50uA).

    

This behavior is confirmed in the output current sink graph, 



An opamp with rail-to-rail outputs would solve the problem. If you want to continue using the LM324 then you have a few choices:-


Provide the LM324 with a negative supply of at least -0.6V (might not be practicable if you don't already have a suitable negative supply). 
Connect a load to the output which pulls down to Ground, eg. a 150&ohm; resistor (probably still won't get the voltage low enough).  
AC couple the op amp's output to the input of the A/D converter, and use a voltage divider to set its bias point at 1.5V. To maintain accuracy the input impedance of the MCP3202 needs to be 1K or less, so a voltage divider using two 1.8k resistors should be OK. You will also have to change R1/R2 so that the LM324 is biased at >=2.1V for sufficient output voltage swing.