Well, the first thing to do is double check the wiring and assure that you have two distinct circuits. Neither the hots or neutrals on the protected side of the GFCI should come in contact with those from another circuit.

It's possible that the rather strong field from the microwave is getting into the GFCI. Particularly older (and cheaper) models have interference issues. You might try replacing it and if possible, moving it further from the microwave.

The microwave and outlet are both on separate circuits, so I changed out the GFI, and now everything seems to work. It took a while to find the circuit because it was on the refrigerator circuit, and I thought a refrigerator had to be on its own circuit. Oh well, it work now. Thanks for your help.

No, there is no requirement that the refrigerator must be on its own circuit.