Can any woman now sue the state for child support if she lives in a state that prevents abortion?

Here is my thinking. Roe vs. Wade was or is being overturned and from what I understand it will be left up to the states to decide if abortion is illegal in their state. So if a young pregnant woman without the financial means of moving to get an abortion is forced to have a child, can she argue the state should provide for the child financially until it is an adult?
Also if such a case was won, wouldn’t it mean that all women are entitled to have the state support their children since their rights to decide on what happens in their life is being decided by that state?