24-year-old brother refuses to financially provide for his 16-year-old sisters' baby: 'She said I was abandoning her like her boyfriend did.'

This is the best way to go. If he promises to "support her" in a vague way, she might think that he's agreeing to support her the way a father would: every day for the next 18 years. If she's constantly asking him for financial support for every little thing, he might start to resent her, and that wouldn't be good for either of them. She clearly sees him as someone she can rely on for financial support, but not much emotional support. He's obviously a decent brother, but he probably already feels kind of weird about supporting someone who couldn't even call him up to tell her that she was expecting a baby. He's not a huge presence in her life. If he were, he would've been the first person she called. 

If he tells her in no uncertain terms, "I will support your legal fees so that you can get child support, but I won't give you money indefinitely because you haven't sought child support," then she can either take that deal or leave it. She will get child support, even if the father of her child threatens legal action. If they take a DNA test and the baby is his, he won't have a case against it.