It is technically possible. I had the same thought as you and looked into it some time ago. The only references I ever found to it actually being implemented were in cases of giant walk-in (even warehouse sized) refrigerators/freezers in places where district cooling exists.

For something the size of a home fridge the costs would be immense compared to the energy savings. You're far far better off spending that money on more efficient heat pumps and more/better insulation.

The federal limit for the amount of power a fridge can pull is 527 kWh/year, which at my rates (admittedly on the low end these days) is ~$70/year. There are very commonly available fridges that are < 300 kWh/year, which would be ~< $40/year. So before even taking into account the efficiencies you implicitly make back when you're heating your house anyway, that's your per-attached-appliance limit for input cost on building out, maintaining and running that system.

Though, I admit that the idea of a completely silent fridge & A/C is alluring. Here's to hoping we make some breakthroughs on sold-state heat pumps.