Am I sinful if I stay away from my sick sister who denies my help to her, and does religion oblige me to stay by her side and support her?

You must maintain good relations with your sister. It is not permissible to cut ties with her, as severing kinship ties is a major sin. The quarrels that emanate from her do not justify cutting ties. Rather, maintain relations with her to the extent that it protects you from her harm. Maintaining ties with her does not necessitate her care, service, or nursing unless she is poor, in which case her sustenance and care would be the responsibility of whoever is obligated to support her among her heirs. The kinship tie required by Islamic law is fulfilled by anything considered a bond in custom, such as financial support, fulfilling needs, visiting, and exchanging greetings through correspondence.