Been troubleshooting this for a week now. After 3 years working perfectly (age of fridge) I noticed the fridge had no power last weekend. The GFCI had tripped on one of two circuits that power my kitchen. I reset it and it tripped again instantly. I got a 12ga extension cord and plugged it into the other gfci protected circuit and it has been working fine. The troubled circuit consists of the line from the breaker to a to a 20a gfci and then runs to two consecutive standard outlets. The fridge is the only device plugged into the circuit. I replaced the gfci and instead of tripping instantly, it now trips every 5 to 10 minutes. Seems to coincide with the cooling or defrost cycle. Tested the fridge on several other gfci protected circuits in the house with no issue. Testing each of the 3 outlets on my troubled circuit shows no faults and I checked all the wiring I can see. I’m out of ideas!
As per code:
GFCI protection is only required for kitchen receptacles that serve the countertop surfaces. There's no requirement to GFCI protect receptacles that serve a refrigerator. Unless the fridge is plugged into a countertop receptacle.
Why does the Fridge trips the GFCI?
Any inductive load when switched off, can produce electromagnetic interference (EMI). This interference can, and often does, trip GFCI devices. Fridge compressors have a few inductive loads, that could cause the trip. To prevent that most modern frige have EMI filter.
As far as a fix, dedicated circuit is not, technically, the solution. You can have the refrigerator on a circuit that is part of the rest of the kitchen, just make it not be GFCI protected. The key - dedicated or not - is to place the receptacle in a location where (a) it is more than 6' from the edge of any sink and (b) not in a position to serve the counter.
Arguably, putting the receptacle behind the refrigerator designates it as "not for the counter", but using a single receptacle instead of a typical duplex receptacle will make it 100% clear. A single receptacle may cost more than a duplex, but don't worry because (a) the cost is still very low (at Home Depot currently < $4 for a Leviton) and (b) they are usually good quality (e.g., "Commercial Grade").
As far as dedicated or not, if you are using (as in the original post) GFCI/receptacle rather than GFCI/breaker, simply pigtail the wires for the single refrigerator receptacle off of the LINE side of a GFCI/receptacle rather than the LOAD side and you're all set.
Note that if this is a dedicated circuit with a single receptacle then if it is a 20A circuit then it must be a 20A receptacle. A 20A circuit with multiple receptacles (even just one duplex receptacle) can use all 15A, all 20A or a mixture. Kitchen circuits serving the countertop receptacles must be 20A, so if it is not dedicated and the other receptacles are countertop receptacles then it must be a 20A circuit. If it is dedicated then I see no reason it couldn't be 15A instead of 20A - same price breaker, but save a little using 14 AWG wire.