All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
Indeed, you did well by supplicating Allaah to guide your husband and enable him to repent and you are right in clarifying to him that what he is doing is wrong. However, he was wrong by telling you that this matter is between him and his Lord and that you have nothing to do with it. Rather, a Muslim is religiously obliged to enjoin good and forbid evil. Abu Sa’eed Al-Khudri narrated: “I heard the Prophet saying: ''Whoever amongst you sees an evil, he should change it with his hand (i.e. physically), if he is unable, then he should change it with his tongue (i.e. verbally) and if he is unable, then he should change it with his heart (i.e. by hating it in his heart) and that is the weakest level of faith." [Muslim]
If it is confirmed that he establishes illicit relationships with other women, then there is no doubt that he is committing a great evil. However, it is not permissible for you to accuse him of Zina (adultery) with them unless this is confirmed with absolute evidence so that you would not be accusing him of Zina falsely. Indeed, there is a severe threat about accusing someone of Zina; for more benefit, please refer to Fatwa 85285.
As regards you going out of the home without his permission because of him committing sins, then this is not permissible. Also, him committing sins is not a sound reason for you not to obey him within what is permissible. Therefore, you must go back to your marital home; for more benefit on a wife going out of the home without her husband’s permission without a necessity, please refer to Fatwa 91788.
Furthermore, you are not obligated to ask him for divorce, but it is desirable for you to do so as long as he insists on doing what is forbidden because it is desirable for a wife to ask for divorce from a dissolute husband who does not observe the rights of Allaah. Also, if you wish to be patient and continue to stay with him and advise him hoping that Allaah will guide him, then you may do so.
Allaah Knows best.