All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
We have already issued Fatwa 83830 clarifying that Islam has forbidden harming animals without right and it is confirmed that the Prophet said that a woman entered Hellfire because of a cat which she tied until it died.
Therefore, it is not permissible for that man to kill that cat and whoever does that is sinful whether he is a Muslim or a non-Muslim because it is forbidden to kill cats or other non-harmful animals without any sound reason. The Prophet said: “Whoever kills a sparrow or anything bigger than that without a just cause, Allaah will hold him accountable (on the Day of Judgment). He was asked: “O Messenger of Allaah, what is a just cause?” He replied: “That he will kill it to eat, not simply to chop off its head and throw it away.” [An-Nassa’ee]
Even if we presume that that man was harmed by this cat, he should have asked you to repel its harm from him but not resort to killing it.
Finally, we advise you to endeavor to forget this cat and forgive that man while advising him not to do this again as this act is contrary to the teachings of Islam.
Allaah Knows best.