Thanks for everyone's advice. I am making progress.
Yesterday I rewired the GFCI above the sink. I now have the power coming into it from the panel to "line", power going out to "load" goes on to another receptical. I have the refrigerator fed from the "line" side as well so that it is not affected by the GFCI. After doing this, the refrigerator was turned on and the GFCI does not trip.
Did I do something wrong by running the wire off the "line" side to the refrigerator? There was no room in the box to pigtail.
Also, I turned the refrigerator on for 30 minutes and it does not seem to be working on electric. Would the most likely problem be a heating element? Today I'll try to light it using propane and see if it works on propane.
Thanks for your help.