How do I fix a shutoff valve that isn't opening?
From the way you describe it, it sounds like it's a regular gate valve.  You might be able to repair it as you describe by removing the packing nut and replacing whatever is broken or stripped out inside.  

The only exception to that would be a ball type valve, but those will just have a handle that makes only a 1/4 turn.  Those generally aren't repairable, but then again they almost never fail either.

Personally, I would just remove and replace the gate valve with a good ball valve.  Gate valves have a tendency to rot out or seize up from lack of use.  I actually had one once that disintegrated in my hand when I tried to turn it off (spraying water everywhere and making a nice mess).  Ball valves are just much more reliable.