I say you can do it.....
Hi people - question: If I supply water to refrigerator with a tee from the wash machine hose faucet/outlet coming out of wall and then hook up wash hose back in series to washer will it possibly
stink up water to fridge? could it actually backflow into fridge supply? Will a check valve be needed?
If it stinks up the frig then you have done something wrong. or you have another problem
Logically, your connection will be no different than if you put a tee in the supply line so it will work. One thing you might not have considered is that if you shut the valve to the washer off, this supply line will also be cut off. Most ice maker connections tee from a supply line and have their own shut off valve.