Can I disconnect a dishwasher from a GFI and leave the circuit otherwise intact?
So your Dish washer is wired through as a load or wired in parallel and down line from the GFCI ? 

Just curious your reason why you want to do this? Perhaps there might be another issue you have and your resolution might need some additional information so you can accomplish all your goals correctly ? 

Now after those questions the answers based on just disconnecting the washer.

Yes you can un-wire the dish washer from the GFCI as long as nothing unusual was done in the initial wiring. In other words the GFCI must be up-line from the dish washer as one would expect in a proper installation.

If the installation was not proper you can still do what you want, but will need to rework your wiring to the GFCI.