Let's talk about safe sex in Europe. It's quite the big legal subject, but one needn't get into all the details.
I'll start with mine: Sweden
Fatherhood carries the responsibility for the childs livelihood, same as motherhood. A father who doesn't live in the same household as his child and/or isn't legal guardian will have to pay money to the child (or usually mother) until they are 18 or 21, depending on if they are still going to school. If the father does not pay, the government will pay instead, and then collect the money from him.
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If a married woman is pregnant, the husband is legally the father. It doesn't matter if somebody else knocked her up. If a married man wants to challenge his fatherhood of a child, he will in practice need the cooperation of the actual father to prove that he isn't.
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If an unmarried woman is pregnant, a man can sign up for fatherhood. If a man doesn't, social services will have to examine who the father is. This can be done through various methods and if they fail to ascertain who the father is, but the pregnant woman says who it is, it will be taken to court. In court, as far as I know, the burden of proof is nearly non-existant.
In all cases, if a man is legally determined to be the father of a child, but disputes this, or suspects he isn't, he will need the actual father to help determine that he isn't.
What's it like in your country?
Wow, so paternity determined by DNA is insufficient to determine a child isn't a man's progeny in Sweden? Further, there is no recourse in the event of the wife's infidelity without the actual father stepping up? That feels like an unjust trap.
Towards your actual question, in the US it's not automatic. Child support is almost always a consideration for adjudicating divorce if minor children are present. Unwed mothers would have to seek support through the courts if the father refuses.
Does bring up the question of paternal rights in cases where the man does not want a child. He can't have an abortion like the woman if she doesn't want the pregnancy. Definitely a hard thing to balance and be just in all cases.
Sorry, I may have made an error or been unclear here. A man can ask for DNA testing if he thinks the child isn't his. However, the mother (child actually, but spoken for by the mother) don't have to cooperate with that, which means that if they are married, he'd still have legal fatherhood.
That's really fucking stupid lol