Start by making sure the black fill line above the ice maker is not frozen. If it is, then that is your problem. The water valve consists of two valves, one for the dispenser and one for the ice maker so yes, one side can be bad. It is more likely the water line if frozen or the ice maker isn't working properly.

When you look at the ice maker, there are some fingers that move to harvest the ice and they should be at about the 2 O'Clock position, or pointed toward your forehead as you look at them. Make sure they are in the correct position and are not loose when you try and move them. Also, the red light on the right side of the freezer liner should be flashing a rhythmic heartbeat when the door is opened. As long as the water line is not frozen at the end, the ice maker module is the likely failure, but the control board can also be the failure. Check the stripper fingers and the light first. Then try removing the three screws that hold the panel on the right side which will remove the control board. Using an insulated paper clip, jump between the black, and black/white wires to bypass the electronics. There will be 120vac so don't touch the clip with your fingers. If the ice maker starts to work, you need a new emitter and receiver boards. If it doesn't, then we are back to the ice maker module.