I was told he could sue me if I don't let him know the baby was born.
Who told you that this man could sue you? Under what legal theory is this creep planning to sue you?
Frankly, madam, I think this bum is trying to bully you, scare you, and/or control you?
The only one that eventually could be sued is the "UNKNOWN baby daddy" for CHILD SUPPORT to the KNOWN mother.
Madam, I hope you deliver a healthy, happy, smart, perky, beautiful baby.
You are under no legal obligation to reveal anything to anyone, unless a court orders you to do so.
By the way, no court would order you to tell a man he's a father, because you're not a paternity testing company.
As I said, ignore this bully. If he harasses you, I suggest you get a restraining order, or file a police report if he's stalking, threatening, or bullying you.