Praise be to Allah.
I ask Allaah to reward you greatly for your keenness to avoid sin. Allaah, may He be glorified and exalted, loves a pious slave who remembers that He is watching in secret and in the open, and takes stock of himself in all his affairs, and does not rest until he is assured that he has not disobeyed Allaah or earned His wrath.
With regard to what you have mentioned about your disagreement with the wife of one of your relatives, you have taken the right first step by greeting her on every occasion. You should complete the good deed by visiting her, even if it is only once, so that there will not be any resentment or grudge left in the heart. By doing this you will cut off a way by which the Shaytaan could reach you. Strive to keep away from anything that could provoke disagreements and arguments between you, because it is not permissible for a Muslim to argue with his brother over trivial worldly matters. It is mustahabb for the one who has been wronged to forgive his brother, so what about a dispute that seems to be about ordinary things?
The wife of a relative is not a mahram, but upholding ties with this relative – if he is a mahram – includes visiting his wife, asking how she is, and taking an interest in her affairs.
In Fataawa al-Lajnah al-Daa’imah it says (26/127):
Islam enjoins instilling love among the Muslims. It urges them to love one another, be compassionate towards one another, and uphold ties with one another so that their affairs will be set right, and they will be united against others. Islam warns them against enmity and hatred, and forbids them to forsake one another or cut off ties with one another. Forsaking a person for more than three days is haraam.
In al-Saheehayn and elsewhere it is narrated from Abu Ayyoob al-Ansaari (may Allaah be pleased with him) that the Messenger of Allaah (peace and blessings of Allaah be upon him) said: “It is not permissible for a Muslim to forsake his brother for more than three days, each of them turning away from the other if they meet. The better of them is the one who greets the other first.”
In Sunan al-Tirmidhi it is narrated from Abu Hurayrah (may Allaah be pleased with him) that the Messenger of Allaah (peace and blessings of Allaah be upon him) said: “Beware of conflict, for it is the shaver” i.e., it erases religious commitment.
What the Muslim must do if there is a disagreement between him and his brother is to go to him and greet him with salaam, and treat him kindly so as to reconcile between them. There is great reward in that, and avoidance of sin. End quote.
Conclusion: you are not regarded as severing family ties because of this, but you should strive to resolve the differences between you.
We ask Allaah to set the Muslims’ affairs straight.
And Allaah knows best.