[villagetinker]

There is another possibility. In some cases, the wiring uses what is called a "shared neural". In this case, there is a 4 wire cable (black, white, RED, and bare copper ground), The black goes to one circuit breaker, the RED goes to another circuit breaker, and the white neutral is SHARED between the 2 circuits. If you have this wiring, I am sure the "shared neutral" will interfere with the operation of the GFI receptacle.
The breakers in the circuit breaker box should be labeled. I would turn off the one labeled for the refrigerator, and see what is NOT operating. Then I would turn it back on make sure everything is working and then turn of the circuit breaker supplying the GFI, and see what is not working. If you are comfortable taking the cover off the circuit breaker panel, you can also check for a RED wire on either of the 2 circuit breakers you just operated (shared neutral).
I would also check the wiring on the GFI to make sure it is correct, both of the wires from the HOT cable need to be n the SAME side of the GFI usually labeled , LINE, ad both wires going to the downstream load need to b one the same side of the GFI, usually labeled LOAD.
Hope this helps.