In cali, children born in the course of a marriage are presumed to be the child of both parents. The presumption is extremely strong and can rarely be disturbed. For example, wife cheats, has another man's child, DNA shows it. The biological father cannot petition the court for custody or visitation, etc. It is the husband's kid. The husband though may be able to challenge the paternity under certain circumstances.
But here, the spouse volunteered and knew. She's not getting out of paying.