Is it possible to shorten a chain? I have the following chain/gear setup:


I like to get rid of the rear wheel gears. However I don't want to buy any new parts. That is why I am wondering if I simply can remove all the parts from the rear gear shift, and simply shorten the chain?
NO

The derailleur holds chain tension.  

At a minimum your would need a chain tensioner.
And it would need to line up with A gear. 

Single speed and fixie have horizontal dropout to adjust tension.