Most of the time when a valve is not seating properly and lets water through in the off position there is a house related water pressure issue. It is possible that you could have a valve allowing water to pee past the valve gate with full house water pressure but that valve would have been in service for a long time or the water supply is extremely hard or contaminated. This is a common problem when the ice maker water supply line is connected to the house water line with a piercing valve. The point where the needle pokes a hole in the water line does not always make a full size hole. This will reduce the amount of water / pressure that gets through holding the valve closed. We see this problem more on non city pressure water supplies. Houses with pumps and reverse osmosis systems sometimes have low water pressure.
And by the way I have seen this problem many times when people have a second home or a cabin that they only use periodically. To prevent a catastrophe they turn the main water supply off to the cabin. Well if that valve does not completely shut off, it will allow just a small amount of water to pee through. That small amount of water is not sufficient to keep water valves closed on the ice maker, the washing machine, or the dishwasher. So it’s not uncommon to see any one of these appliances fill with water when no one is home. If that happens on the refrigerator valve and they leave it running, the tube is going to freeze. You can easily prevent this issue if you turn the sink water faucets on in the kitchen when you turn the main water supply off. This will give an easy way out for any water that happens to be seeping through the main water valve shut off.
The best way to verify that you have sufficient water pressure to the refrigerator valve is to turn the main water supply off to the house. Do not turn off the piercing valve attached to the waterline going to the icemaker. That’s because you may break loose any obstruction and may not be able to diagnose the problem. Once you have the main water supply turned off disconnect the water from the ice maker water valve. Use a 5 gallon bucket or other large container and have someone turn the main water supply back on. Now you can monitor the amount of water pressure coming out of the line. If it is low you can work backwards to correct it.