I'd say your probably screwed without tearing the wall out. Chances are you can't just put a wrench on that valve and remove it because you can't get to the back side to hold back on, if it's just screwed into a pipe. Or if it's like the pictures, you would have to cut most of the wall out to get to the pipes. I had a similar issue with the washing machine box, the gate valves won't shut off completely, but they don't leak. My solution was to put a nice new ball valve on my main water line in the basement. If I need to change my washing machine hoses, I can turn the main off easy and quick. It's also in a good spot and well labeled so if the wife or kids have any kind of issue with water or a leak and I'm not around, they can easily turn the water off to the house. As long as your valve isn't leaking, I'd leave well enough alone. You aren't going to go back there and turn off the ice maker daily, weekly, or even monthly. Probably only every 10 years when you replace a refrigerator.