Assalaamu alaykum. I have a question about these two hadiths:
“The Muslim is not allowed to abandon a Muslim brother more than three days. If three days pass and he meets him, then he should greet him. If he (the other believer) returns his greeting, then they share the reward; but if he does not return it, then he (alone) will incur the sin.”
“The one who breaks off from his (Muslim) brother for more than three days and dies (during this period) will go to Hell.” I am a Muslim man, and I have completely cut contact with a non-Mahram (permanently unmarriageable) Muslim woman due to some personal reasons (disagreement, dispute, and so on). I know that you should cut off contact with the Muslim woman (for the sake of Allaah) because she is a non-Mahram. But will I go to Hell if I do not reconcile with this Muslim woman (who is a non-Mahram) because I cut off contact with her because of some personal reasons and not for the sake of Allaah? Please answer my question, because it is important for me. May Allaah reward you.

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.

The first hadeeth was reported by Abu Daawood and others, and the scholars criticized its chain of narrators; whereas the second hadeeth was reported by Abu Daawood and others, and it was classified by Al-Albaani as saheeh (sound).

It is forbidden in principle for a Muslim man to desert his fellow Muslim brother for more than three days without a valid reason. This is in regard to two disputers, or a Muslim who meets his Muslim brother and does not speak to him or greet him as a way of deserting him, because the Prophet said, “It is not permissible for a Muslim to desert his fellow Muslim more than three days, they meet each other, but each one turns away from the other, and the best of the two is the one who greets the other first.” [Al-Bukhari and Muslim]

With regard to a woman who is not a Mahram to a man, in principle there is no relation between her and him because she is a non-Mahram to him. Please, refer to fatwa 81356. So it is not permissible for him to talk to her except for a need and within the limits of the Sharia. In case he quarrels with her, it is an obligation to remove the dissension and hatred that prevents the deeds of the two disputing parties from being presented to Allah as well as His forgiveness which every believer gains on Mondays and Thursdays, as it was narrated by Abu Hurayrah, may Allah be pleased with him, that the Prophet said, “The deeds are presented every Monday and Thursday, and on this day Allah forgives every servant who does not associate anything to Allah, except a man who has enmity and disputes with his Muslim brother, and it is said, 'Delay these two until they reconcile, delay these two until they reconcile.'” [Muslim]

Another narration reads, “The doors of Paradise open on Mondays and Thursdays; every servant who does not associate anything to Allah will be forgiven, except a man who has enmity and disputes with his Muslim brother...” [Muslim]

However, you did not clarify the reason why you changed your attitude towards that woman and what made you enemies; in any case, you can remove the enmity and hatred between you without meeting her and without establishing a relationship with her by removing the causes of hatred between you. So you forgive her if she wronged you, or you ask her to forgive you if you had wronged her, or you give her back her right if it is something material that you had taken from her.

If there is no injustice between you and her, then in principle you have nothing to do with her, and you must cut any relationship with her, as we have mentioned. As long as she is a non-Mahram to you, you are not included in the threat mentioned in the hadeeth.

The deserting that is Islamically forbidden is the one in which the person intends to cut relations with someone. As for not greeting a person because of not meeting him, or when a man does not greet a non-Mahram woman because of fearing to be tempted by her, then this is not the forbidden deserting.

If you desert a woman for any other purpose than the sake of Allah, then you will not be rewarded for doing so, because deeds are judged according to their intentions, and every person will get what he had intended.

Nonetheless, there is no harm in greeting this woman or any other woman, or responding to her greeting when she greets you, if there is no fear of temptation. The scholars stated that it is permissible to greet a non-Mahram woman if one does not fear to be tempted by her. An-Nawawi said in Riyaadh As-Saaliheen, “Chapter of a man greeting a non-Mahram woman or non-Mahram women when he does not fear to be tempted by them.”

They also stated that deserting ends by just greeting (saying Salaam) someone or by responding to his greeting (with Salaam); Fath Al-Baari reads, “Most scholars said that deserting ends by just greeting with Salaam or responding to a greeting (of Salaam).”

Allah knows best.