Why do deraillers require a freewheel? I've seen fixies with internal hubs, but a non-freewheeling rear wheel with deraillers seems to be unrepresented:

            |        Freewheel?        |
Derailler?  |      Yes      |    No    |  
      Yes   | Standard bike |    ??    |
       No   | Single-speed  |  Fixie   |


Practicality aside, is there a mechanical constraint involved here? What happens if you try it?
For the same reason you can't have a fixie with a chain tensioner.

The load on the chain when slowing a fixie is too great and in the wrong direction for a derailler or tensioner to hold. The cage will be pulled forward and your chain will skip make a nasty noise and most likely come off or break something expensive.