Yes, dogs do show remorse which indicates a sophisticated level of social interaction. The question needs fully explaining, however. There are two aspects to it.
The dog is fearful not remorseful
When a dog does something wrong and knows it and if they are in the presence of their owner, they are able to detect the slight changes in body language of their owner as a prelude to being reprimanded. The dog becomes submissive and fearful but this is interpreted as remorse by their owner from the expression on their dog’s face as he approaches submissively before being reprimanded. They can guess what is to come.
The dog is remorseful
Dog owners say that their dogs behave submissively and remorseful even before their owner has discovered their misdeed. A dog has learnt from past experiences that they have behaved badly by chewing up something due to boredom and he may greet his owner in a friendly and submissive way. The owner has not seen the damage to his possessions. Therefore he cannot give those early warning clues about impending reprimands to their dog, that I refer to above. Therefore the dog’s behaviour is an independent appeasement action knowing that what he’s done is wrong by human rules. This in effect means that the dog can show remorse.
The same sort of behaviour has been seen in wolves. A good example might illustrate the point. In a wolfpack there are dominant and submissive animals. Let’s say someone throws some meat into the pack and a submissive animal takes it. Under wolfpack rules of social interaction, that animal has a right to keep the meat despite the hierarchy. The dominant wolf knows this. By chance the dominant animal manages to take the meat and eat it while maintaining what is called the “ownership zone” which is 12 inches around a wolf’s nose. Later on the submissive wolf will go to the more dominant one and apologise for breaking the rules of the pack and in doing so provide a clear message that they were not trying to usurp the dominant wolf’s position as pack leader. This is a form of appeasement within a wolfpack under certain circumstances.