Does my not speaking to my brother or greeting him, and shunning him because of his committing sins, his refusal of advice, and his aggression toward me, constitute a severing of the ties of kinship that necessitates sin?

The fundamental principle is the preservation of affection and connection among Muslims. Abandonment (hajr) and estrangement (qati'ah) are not ends in themselves but rather means. Therefore, consider your brother's situation: if abandoning him serves as a deterrent to his persistence in sin, then there is no harm in it; in fact, it is desirable, and this abandonment should be for the sake of Allah. However, if abandonment does not achieve a legitimate religious purpose, then advice and maintaining affection take precedence over abandonment that yields no benefit. And if abandoning this brother repels a feared harm—such as his striking or reviling—then there is no harm in it.