Is the woman considered sinful for severing ties with her relative who falsely accuses her of committing indecencies, despite her attempts to clear her name and his continued insistence on his opinion, such that she rarely speaks to him and feels hatred towards him? Will her good deeds be halted because of this?

If your relative has accused you of wrongdoing without proof, he has wronged you, and he must repent and seek your forgiveness for this injustice. It is not permissible to cut off this relative if he is among the kin whose ties must be maintained, merely because of his wrongdoing. Rather, kinship should be maintained to the extent that it does not cause harm. As it is stated in the Hadith (definition): "Be generous with your bounty and maintain ties with them; for there will always be a supporter from Allah with you." Abandoning a relative for one's own personal gratification exposes one to the warning mentioned in the Hadith: "…except for a person between whom and his brother there is animosity, then it is said: 'Leave these two until they reconcile.'" However, if the boycott is for a legitimate Sharia (definition)-compliant reason, such as to rebuke him for a sin, then it does not fall under this warning.