Sorry, going to post a wall of text and going to class, bbl.
This probably varies from country to country, and maybe even state to state in the US, but generally it's the mother who has all the parental rights over an out of marriage child.
As such the catholic woman would be able to give the child for adoption, ending all her responsibilities and rights over the child. In the second case the woman will also be able to give the child for adoption. If the father wants the child, he might have to go about it the hard way, the child can be ruled into third party custody if he's not deemed to be the better option for the child's upbringing.
Marriage is a different beast because it was obviously never designed for anything having to do with fairness and choice. However if the catholic woman was married to the man and didn't want to get pregnant, she WOULD be still able to get an abortion without consulting him, in a lot of countries that allow abortion. Yes, that would be against her faith, but at this point this gets very conflicting when considering the Catholic interpretation of sex and marriage, and is ultimately a dispute about ideology rather than rights.
I actually don't think that men's and womens rights should be put absolutely equal when it comes to pregnancy. At least not until science is up to having a complete artificial womb birth that would even out the actual footing when it comes to inconveniences of pregnancy and birth, economical, social or personal, and both parents can be just donors. Just swapping the sexes simply cannot currently make for a valid comparison because men aren't getting pregnant and outside of marriage don't have the parental rights by default.
But when it comes to child support, the interpretation of laws that end up like the two presented cases, disturbs me. A case with circumstances allowing for a swap of sexes and still the same situation and ruling would still disturb me. This is about a law being rigidly interpretated in view of both parties' ethical treatment of the other, measures against and for pregnancy being taken and ultimately the distribution of responsibility for doing something.
Given the current possibilities for both men and women to decide pro and con having a child, I think laws regarding those decisions should allow for a lot more variation. And it's depressing that we have some profit-oriented laws (e.g. copyright) include all sorts of subclauses and differentiations to cover as much ground as they can, but a basic social law is stuck on a binary system.
We will have double standards for a long time, they can go both ways, and there are a lot that work against women, we've had a patriarchal society at least since the agricultural revolution. They won't be done away with making sides either scared or empowered, especially not when it comes to children. We still want people to have children and families, feel confident and comfortable with the (level of) commitment they have decided on. And while it was easy in the past for men to just get a wife, have kids and not care about what the wife had to say very much, and it was a given for women that their values would lie entirely on getting married and being said wife, this isn't the case anymore, and now there needs to be proper communication and a proper shift of responsibilities.
Regarding the issue of joint property that sapphy made. Personally I very much dislike the traditional roles for both sexes, as well as the value society tends to attribute to their traditional occupations. Household work, raising your kids, providing healthy food on a daily basis, providing domestic comfort for a family's well-being - this isn't even considered economic activity in the context of the family, yet it is considered so when responsibility is given to a third party. It is actually work, a lot of which we will likely not be able to replace by machines any time soon, while a lot of the traditionally worthy, hard and technically-wise complex jobs are successfully being done by robots. Taking all those traditionally female occupations and duties out of a family would completely ruin this family, and in the long run drive down society. IMO it's largely already happening. I can see no good about downplaying the importance of those occupations by measuring it by money earned. The focus when looking for equality should be for both men and women to cover between them the full range of all duties, in a split agreed upon, not just match or split the earnings and in the process do the bare minimum of the rest. Stay single and childless, make a career and keep all the money if that is the actual worth of a family and of life.