What is the ruling on someone who curses another person due to their bad manners and ill-treatment, knowing that the one who cursed heard his curse? And how can one atone for this cursing if it was an injustice?

The basic principle is the impermissibility of slandering and backbiting a Muslim, except for a valid reason. If there is no valid reason, then your father must repent to Allah with remorse, cessation, and a firm resolve not to return to it. He must also seek forgiveness from those he spoke about, if they heard it. If the backbiting did not reach the backbitten, then remorse and seeking forgiveness are sufficient. But if it reached them, he must seek their forgiveness. If it is impossible to do so due to death or absence, then he should seek Allah’s forgiveness.