The broadcaster said the tweet was “supposed to be a joke.”
Duke and Duchess of Sussex named their son Archie Harrison Mountbatten-Windsor.
The infant was born early Monday morning.
“Harry and Meghan: Becoming Royal” will premiere on May 27.
The Duke and Duchess of Sussex’s baby is due next spring.
Nearly 30 million Americans tuned in to the ceremony.
Thomas Markle says he is recovering from heart surgery.
Prince Harry and Meghan Markle exchanged vows Saturday during a star-studded ceremony.
“I have always cared for my father and hope he can be given the space he needs,” she said.
Kensington Palace has called the matter a “difficult situation.”