What do you do when one dog dies and the other doesn't act like itself anymore?

I agree that your dog is grieving over it's friend and doesn't understand why the other dog is no longer there. Some get over it quickly others don't. I think perhaps another dog introduced into the family might help but I would ask my vet his opinion first before getting one. Try enticing your dog with some table food just to get some nourishment into him. We had to put an aging cow down once who was so feeble with age that she could no longer get up when she layed down. When we had her put down it was in the pasture with the rest of the heard and they stood and balled for several days at the spot where she died.Then they went on about their business.