Won't make any difference if he signs the birth certificate or not, should they choose to "get him in trouble". There's DNA, ya know?
Actually, he has gotten himself in trouble, so let's put the blame where it belongs.
But, no....so long as things are hunky dorey with the girl and her parents, then no legal agency is going to step forward and file charges. Hell, they won't even know about it.
But if things go south with that girl or her family, he can kiss his ^^^ goodbye.