Robbie added: "Baby, Mummy and Daddy are all rockin.... Thank you for your best wishes XXXX."
Teddy is the first child for the couple, who married two years ago, with their wedding pictures appearing exclusively in HELLO! magazine.
Robbie recently voiced his concerns that he would be on stage when the baby made her entrance.
"I've had a bit of a falling out with the wife," he told DJ Chris Evans.
"I've got gigs this week, the baby could come at any minute, the wife has said, 'If I go into labour and you've got a gig in say Leeds or Dublin, you've got to cancel it and come home'."
He also revealed the couple were happy with the traditional name they had chosen.
"Right now I'm ecstatic. We do have a name but we're not going to get it out in case it gets tweeted.
"And as a celebrity we're supposed to call them Aubergine or something like that aren't we?
"I do have a sort of celebrity name but it's not Apple-esque. It's a very solid, old-school name."