From what I've read, that valve has the tubing attached, and is an accident waiting to happen. Not sure how it comes off, but highly suggest you replace the whole thing, valve and supply line. One of the pros that deals with these should be able to tell you how it comes off. You may need to cut it off. Do you know what kind of supply pipe it is attached to (cpvc, copper, pex?)
On mine, I have what amounts to probably 10' of copper tubing in a big coil behind the frig. It's long enough to pull out like a spring if I need to pull the thing out for service. If you want a hose, get a reinforced SS one.