Have you unplugged the refrigerator's 120 VAC cord-- try that (outside access door). If the GFI is tripping because of a problem in the refrigerator, this should identify/eliminate the issue.
If that doesn't cause it to be resettable and stay reset, unplug everything that is plugged into either the GFI or any outlet downstream of it. See if it will stay reset. If not, you have either a wiring problem or if a GFI trips enough, it may not reset.
And, yes, refrigerator 120 VAC heating elements are one of the most common causes of GFI failure.