No. Hate Speech in itself cannot kill. However, it is the actions in which the person who hears the speech that can kill.
There is a saying: "Sticks and stones may break my bones, but words will never break me."
In a sense, this saying is partly true, partly false. Words cannot have a physical impact in the beginning stages, but has more of an emotional charge and a mental impact. Depending on the words, if it is positive and from a trusted source, it generally gives a rush of ecstasy that you have done well. However, on the other hand, if it is from an untrusted source, you may be thankful, but still on guard. It cannot be said the same for the opposite. Negative words denoted from a trusted source may cause a grain of salt, but would be looked upon for consideration, while an untrusted source will be untrusted in the information provided already in the source, rather than straight to the statement. Remember, if you trust a source, you rarely question whether or not the event is true or not, you just take their word for it (hence, trust). But if you do not trust, you will research and prod to see if what they are saying is reasonably true or not.
In this case, we are talking about hate speech. Hate speech as defined by Oxford Languages is "abusive or threatening speech or writing that expresses prejudice against a particular group...". Let us look at it from a couple of perspectives:
1) The Hate Speech is aimed at a third party, and is given towards the general public. This hate speech calls for a violent effort to destroy the group.
Yes. In this case, there is not only a mental attack, but also a physical and emotional attack. The violent effort implies that there will be beatings, robberies, killings, and all the physical violence that comes with it. This part is a definite "kill". The next part is the psychological and mental part. Those who survive will live and remember this event. Typically, it will draw one of two reasonings from them. The First one, is fear, typically for one's own life. Seeing the outcome for their family members, and then continuing to see the government to be in power will lead to the fear for one's own life. The second reasoning is revenge & hatred. This one comes in two parts. The hatred of the group is inevitable as a human, however, the revenge can come in different ways. Some people will take physical revenge in attacking the source. Others may call for help (psychological revenge) and use a typically 3rd party to exact their revenge. The last one, and the hardest, is to forgive.
2) The Hate Speech is aimed from one party to another directly.
This generally can lead to killings, especially with groups that are weaponized and war mongering. Examples can typically be found in warring, unstable states, gangs, and criminal groups. The leaders typically use hate speech in a way to degrade their enemies, which could lead too two things: A blood lust of the enemy to enact revenge on what is said, or an exchange of words that can still lead to physical violence. In this case, both parties are armed, ready and willing to fight, and have a large ego that must be defended when disgraced. This type of hate speech can lead to killing.
3) Hate Speech in the general public.
Whether as a joke or not, we can all agree that there are hate speech in the public, whether it is more apparent in some areas or not. As protected by the First Amendment of Free Speech, this type of Hate Speech does not account for killings. However, that is not too say that it holds the potential for murders. Speech can be defined as articulated sounds that are expressed through the mouth, however, we must understand that there is other ways to show one's emotions. Words typically leads to actions, and it may force those who feel either pressured or 'attack' to physically strike or argue back. In these cases, when it escalates, there is a high probability of somebody dying, and a even higher probability of damage and wounds.
Remember, hate speech in itself is simply words spoken from the mouth. However, the actions in which someone takes when hearing these words are what can kill. Hate speech in itself does not kill.
