Are military members required by regulation to care for their unborn children?

Morally obligated, but we know how that goes. Once the child is born, the mother can go through the court order process. But in 3rd world countries the odds of that are extremely small to non-existent. The more industrialized the country, the better chance of the mother being able to get "non-voluntary" support.
Unfortunately many SM's "hit it and quit it" while overseas and don't give a crap about the child they leave behind. It's not their "problem". On the other hand, there are some local girls who will try to get pregnant by an American SM in an effort to get to "the land of milk and honey".

No. Because, until the child is proven to be his thru DNA testing, he is not obligated to do anything.

No one is legally civilian or military-wise obligated to care for an embryo or a fetus. Until there is an actual child born, there is nothing to care for. I don't know how laws work in other countries, though. That might be something that this person needs to look into but I would bet it's generally the same.

The way it works here in the US: Unmarried males have no rights to a child until they establish their paternity in court. That means they go file to establish paternity, take a DNA test. If it comes back as a match, they will get their rights granted as well as child support. Then the unmarried male can file for custody or visitation. Unmarried females have 100% custody rights of any child they give birth too because it's obvious they are the mother but it's not certain who the father could be.

Married couples - however - are different. If a married woman gives birth while married to her husband that man is automatically presumed to be the legal father. I believe some states you can contest it within like 60 days of birth.

This NCO needs to not worry until there is actually a child or if after birth she comes after him. Since this woman is overseas, if he's not still overseas and she isn't coming after him he might as well just move along and not even attempt to go to court. Unless he wants to know if it's his kid or not - but if he does take this to court he might be asking for a headache.