All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallaahu ʻalayhi wa sallam, is His slave and Messenger.
It is incumbent on Muslims to shun prejudice and fanaticism and the habits of the pre-Islamic era of ignorance. Allah, the Exalted, has blessed us with Islam. All Muslims, in different parts of the world, are bound by the bonds of brotherly love. The only criterion of superiority in the sight of Allah is piety; Allah, the Exalted, says (what means): {Indeed, the noblest of you in the sight of Allah is the most righteous of you.} [Quran 49:13]
In general, if a Muslim curses and insults others out of prejudice and fanaticism, he should be given advice, and the prohibition of what he is doing should be clarified to him. As for that particular incident, we do not fully understand the nature of what that person wrote or his purpose of it and whether his intention was to insult or to forbid that evil.
In any case, if someone insults you or your family and falsely accuses them of what they did not say or do, it is incumbent on you to forbid him from committing this sin, point out his mistake, and give him advice for the sake of Allah. In fact, the insults mentioned in the question have nothing to do with being a Dayyooth, which was defined by the Prophet, sallallaahu ʻalayhi wa sallam, as tolerating and accepting indecency and immoral conduct of one's womenfolk. If a Muslim sees one of his womenfolk committing an immoral act and he approves of it and remains silent about it, he is described as a Dayyooth.
As for exchanging curses and insults, then one is obliged to advise his fellow Muslims who do so and forbid them from it. If they refused to heed the advice, then turn away from them, because they are ignorant. Allah, the Exalted, says to His Prophet, sallallaahu ʻalayhi wa sallam, (what means): {...and turn away from the ignorant.} [Quran 7:199] If you insult him by saying to him, 'You are unjust', 'you are impious,' or the like, then there is no harm on you, because Allah, the Exalted, says (what means): {And whoever avenges himself after having been wronged - those have not upon them any cause (for blame).} [Quran 42:41]
Allah knows best.