Which one is more appropriate in the present continuous? She is having a baby or she is giving birth to a baby.
Both are correct, but mean different things.
"She is giving birth to a baby" means she is actually giving birth at this instant. She is in labour and about to/in the process of giving birth.
"She is having a baby" simply means she is pregnant. It does not mean the actual birth process has started. You could say, "She is in hospital having a baby", but that only means she is ABOUT TO give birth, but it has not started yet.