Thanks, Fencepost. I just did a double check and the tubing is milky white (though slightly transparent). It does have a brass compression fitting at the water line and that also has a shut off valve. There's also an inline water filter.
The upper end of the water tube is taped about 2' up the back of the fridge where it goes into a kind of metal housing that screws onto the back of the fridge. It all looks prety secure.
It sounds like it's pretty safe to move things around. Everyone helped allay my concerns. Now all I have to do is rewire the vent hood and move a small cabinet. This will help the kitchen flow so much better!