We live in an era where awareness of domestic violence and sexual abuse is the highest it has ever been. It takes no imagination whatsoever to think of a few scenarios where a woman wouldn't want to tell her husband she was pregnant and wanted an abortion-- and none of them are nice ones. It's a pretty safe bet that any woman who wants to have an abortion and not tell her husband is in a bad marriage, does not live in a happy home, and quite possibly even a dangerous home. A similar principle applies to parental consent laws, neither of which give the woman the benefit of the doubt that she is in a better position to decide whom she can trust than the state. Both laws have the same goal-- to restrict access to legal abortion by granting someone else other than the woman the right to exercise control over her body.
Essentially, the law in Casey is about the state is inserting itself into a woman's privacy as well as her marriage in a truly pernicious way. But I suppose that's OK for anyone who believes that the framer's intent is paramount, after all the framer's never intended for anyone female to be full, equal citizens under the law.
* A talking point that people need to grasp-- requiring a woman to notify her husband gives him the right to try and restrain the wife from proceeding, hence notification is tantamount to seeking permission.