A spokeswoman for the couple made a statement on behalf of the record producer, PA reports.
"Chris would like to thank all the medical staff and everyone who has been involved in Amanda's care," she said.
"Until Amanda is in a position to indicate what she wants to say, there's no more information I can give."
Holden's condition was changed from critical to stable yesterday and she is being treated in an intensive care unit. She is expected to make a full recovery.
It was reported earlier today that Holden and her newborn daughter Hollie Rose Hughes were "fighting for life" after the birth.
> Amanda Holden: 'I might make Simon Cowell godfather to baby daughter'
Watch Michael McIntyre and David Walliams congratulate Amanda Holden on the birth of her baby below: