Removing rigid copper pipe to install dishwasher
You should be able to turn the body of the angle-stop if the nut is held firm with another wrench.  if you turn the body and not the nut - It is less likely to damage the copper tube.  Depending upon how tight the nut was set - will determine how deformed the copper pipe is.  sometimes it is difficult to remove the ferrel.

You have so much oxidation on this valve - I would replace it.
Personally, I never install an appliance onto an old valve. 

It looks like you have enough room to cut off the entire angle stop.  Use a fine toothed metal blade and cut square - then deburr.  A new multi angle stop can then be used with the matching outlets you need.
The braided dish washer supply lines come in different configurations.  I have found that many of the new dishwashers have a large threaded connector instead of the older 1/4" compression.  check yours.

P.S. check inside your pipes for pitting.  Often this oxidation can be a result of iron in water which destroys copper pipe.