What is the ruling on separating from a father who harms his wife and children with beating and verbal abuse, and slanders his wife's honor, despite attempts to advise him?

If the intention behind abandoning the father is to sever ties, then it is not permissible. Rather, it is obligatory to maintain ties with him, honor him, and treat him with kindness, no matter how much he wrongs. However, if the intention is to avoid him to the extent that it repels his harm, then there is no blame. You have done well by advising your father, and you can seek help from his relatives and friends for advice. Also, increase your supplications for his guidance. A father slandering his wife's honor is a serious matter that could lead to an accusation of slander (Qadhf), which is a grave sin. The punishment (hadd) might be carried out against the perpetrator, or he might face severe torment in the Hereafter. As for the father's statement, "You are not my children," if he intends to deny paternity, then it is a serious matter with severe warnings. But if he intends to disavow a bad act, then it does not fall under the warning.