There are two ways to deal with this.
If your neighbour treatens to hurt or kill your cat it is a threat to damage your property and in the UK is a crime (Criminal Damage Act 1971 — 1971 CHAPTER 48). You should call the police and they should interview your neighbour and that may be enough to stop him or her. However, it will obviously create even more tension between you and your neighbour and may make your relationship intolerable.
In the USA I suspect that the law is very similar to the UK and therefore it is certainly a crime to threaten to hurt or kill your cat.
This is because the cat is considered a “chattel” or a possession, no different to any other item possessed by a person. Hurting a cat is tantamount to criminal damage. Therefore in the USA you should call the police if you want to deal with the matter in that way. But below I recommend an alternative way to deal with the matter which may help to keep on reasonable terms with your neighbour if this is possible.
Proactive Steps – Non-criminal
My neighbor threatens to hurt my cat. How can I stop him? Not an untypical question, but, I hope, reasonably rare. The sensible answer is to be proactive and prevent your neighbor making threats. There is little point calling the police after he has hurt your cat. It may bring some satisfaction but the objective is to stop your cat being hurt. Also getting police involved in general is not a good idea, neither is antagonizing your neighbor even more. However, it would obviously be a crime to hurt or kill your cat in all countries in the West. You’ll just have to prove it and that might be tricky.
If your cat is irritating your neighbor either the neighbor hates/dislikes cats and/or your cat is roaming around his garden irritating a fairly normal person. Perhaps your cat is defecating on your neighbor’s garden? A lot of people hate that.
No matter how much a person loves cats and wants their cat to be content by behaving normally, the cat owner has to think of other people and comply with the law.
In America, the answer might be: keep your cat inside as a lot of cat owners do. In Britain the answer might be: if you have a garden of sufficient size, build a cat enclose. Or in both cases, if and when your cat goes out, supervise the trip or put your cat on a leash. Also, it is more socially responsible to pick up feces if he defecates in a public place, although I understand the difficulties of doing that. Your cat should be prevented from going onto your neighbor’s property if he strongly dislikes it.
All of this may sound tough, even impractical, perhaps, but the first call upon a cat caretaker is the cat’s safety and if your cat is unsafe wandering around outside because of threats to his health and welfare something has to be done, proactively.
Also it is important to try and keep on good terms with your neighbors. Neighbor disputes are the worst kind. They make you miserable. There are no winners. It is lose-lose.
In many neighbourhoods people won’t mind a wandering cat. Many people accept it and like it. It creates a more human and friendly feel to the neighbourhood
In this case someone does mind and you can’t take the risk that he is just making an idle threat. People sometimes trap a domestic cat on the premise that the cat is a stray. They take him to the pound or shelter where his lifespan might be only a matter minutes. Or they might put down poison. It is almost impossible to catch people poisoning cats or prevent people from hurting a companion animal. Shooting at cats is another nasty pastime for some people.