What is the ruling on a wife seeking divorce from a husband who is infatuated with women and has relationships with them, given that there is no absolute certainty of adultery having occurred?

If you are certain that your husband has illicit relationships, then you must advise him and remind him of the punishment for those who commit sins in this world and the Hereafter, and threaten to inform your family and his family of his actions. If he repents and returns to Allah, then that is what is desired. But if he insists on continuing these acts, then there is no blame in seeking a divorce from him, and perhaps that would be the necessary solution if you despair of him giving them up.