Introduction: the reason why I am discussing the law in the UK is because in America the answer will be much more complicated because the states make their own law on these sorts of matters. It is almost impossible to comprehensively research these laws on the internet. If a visitor has a specific question concerning a specific state please ask in a comment.
As for the UK (but possibly excluding Northern Ireland and Scotland where there could be slight differences), the law is relatively straightforward if your dog kills or injures a cat. The same would apply if your dog attacked and injured a person or assistance dog but the penalties are harsher.
An interesting story highlights the law. A kitten belonging to a lady crawled under a hole in a fence and found himself in her neighbour’s garden. The neighbour owned a dog, a Jack Russell. The dog attacked the cat and killed him. The police said that no crime had been committed as did the RSPCA.
The RSPCA said that in order for an offence to be committed it would have to be proved that the dog owner had encouraged his dog to attack another animal or failed to take reasonable steps to prevent the attack or allowed their dog to be dangerously out of control in a public place.
The gov.uk website has a page on dogs being dangerously out of control anywhere which appears to extend the RSPCA advice. This means we are referring to dogs out of control in public or private places such as a neighbour’s house or garden or in the dog owner’s home.
A dog would be deemed to be dangerously out of control if it injured someone or made somebody worry that it might injure them. In addition a court might decide that a dog is dangerously out of control if it attacks someone’s animal or if the owner of a companion animal e.g. a cat, believes that their cat could be injured if they tried to stop a dog attacking him/her.
If it went to a criminal court and the dog’s owner (or person in charge of the dog at the time) was convicted of the crime of letting their dog be dangerously out of control he or she could receive an unlimited fine or be sent to prison for up to 6 months or both. In addition the dog owner might not be allowed to own a dog in the future and the dog may be destroyed.
The penalties are more severe when people or assistance dogs are injured.
It seems to me that the matter turns on the intention and behaviour of the dog’s owner. The dog him/herself is obviously innocent and behaving naturally but if the owner is careless or malicious in respect of controlling their dog then they may well be prosecuted in the criminal courts if they allow their dog to kill a cat.
The relevant statutes are, I believe: Dangerous Dogs Act 1991, Dangerous Dogs Act 1989, Dogs Act 1871.
There would also be a potential claim for compensation against the dog owner by the cat owner in the civil courts. Or perhaps the criminal court would make a compensation order.