It seems unlikely, but it depends on your refrigerator, how often you open it, what you plan to keep in it, and how cold it needs to be kept. A refrigerator isn't designed to work like a cooler. The insulation is different and the door opens to the side, which will exchange air faster than a top opening cooler. Have you thought about just getting a well-insulated cooler? Seems like you could do what you're trying without involving the refrigerator and frozen water, but you'll need to experiment some. Some of the newer coolers are fantastic for a couple of days, but it just depends on what you want to keep in it and for how long and how cool it needs to be kept.
Try just putting a gallon of frozen water in your refrigerator and turn it off. My guess would be that the water would be completely thawed overnight.
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