If I meet someone I dislike (and previously had a verbal fight with) everyday, do I have to greet him with salaam each and every day? Or can I greet him only once in every 3 days?

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 is His slave and Messenger.

Greeting someone you meet with Salaam is desirable and not obligatory. However, it is one of the reasons for entering Paradise. The Prophet said: "You will not enter Paradise until you believe, and you will not (truly) believe until you love one another. Shall I not tell you of something which, if you do it, you will love one another? Spread the greetings of Salaam amongst yourselves." [Muslim]

But if in addition to not greeting with Salaam, one does not speak to the person and cuts relations with him, then this is considered as deserting him. An-Nawawi said when clarifying the forbidden type of deserting a Muslim: "The forbidden type of deserting (a Muslim) is to abandon the greeting with Salaam and to turn away (from one another) when meeting…" [End of quote]

It is known that it is not permissible to desert a Muslim more than three days without a legitimate reason for the Hadeeth that reads: "It is not permissible for a Muslim to desert his fellow Muslim more than three days; they meet but each one refuses to speak to the other, and the best of the two will be the one who starts greeting the other first." [Al-Bukhaari and Muslim]

Abul-Waleed Al-Baaji said in Al-Muntaqa: "The Prophet’s saying 'they meet but each one refuses to speak to the other', means -Allah knows best -that each one of them turns away from his fellow Muslim brother as a way of deserting him; so he does not greet him with Salaam and does not talk to him. This is the extent of deserting that he forbade (let alone anything extra)." [End of quote]

Therefore, it is not permissible for you, dear brother, to desert any Muslim brother for more than three days just because of a disagreement or argument between you.

It is also not permissible to completely hate a Muslim: a Muslim is loved for the good that he does and for his Islam, and if he falls into sin or is negligent, then he is hated to this extent only.

Allah knows best.