All good advice so far.
You sure your refrigerator is not the source of the leak in that area? At least something to verify.
The gutters, pull out that rubber/plastic trim material and look for rusted screws holding the gutter on. That can definitely be a source of water intrusion. Water gets between that plastic trim piece and the gutter and will stay wet, rusting screws and letting water in. Remove each screw, shot of sealant and replace with new screw. That, along with the other recommendations about new sealant to fill that gap between the roof material and gutter.