It does not need 'putting down' it needs a responsible owner who will work with and/or muzzle it.
Some dogs kill cats, unfortunately. It's not nice, but it happens. It's in some dog's nature to chase and kill smaller animals. It does not make them a danger to people or children, not unless they're small, four legged and covered in hair and smell like wildlife, in which case I would strongly advise a trip to the GP 
As for the hitting, it's not an ideal way of dealing with the situation, but assuming he was hitting the dog to get it off of the cat, it is understandable. When mine attacked my cat I think I kicked one of them 
Not deliberately I was panicking and needed to keep them off the cat. I wasn't thinking about what I was doing, only about saving the cat (who survived largely unscathed)
I did not consider punishing the dogs afterwards, nor did I blame them. I blamed myself for letting it happen. One of the instigators has now been rehomed (she was a foster, she stayed after the attack until she found her forever home) and the other has been retrained. All dog/cat interaction is now supervised and they are never left alone. I am well aware my terrier would chase and kill a strange cat, given the opportunity so I keep him leashed until I am confident we are a cat free zone. He has not, to this day, killed anyone's pet cat. I don't give him the chance to.