Fabian Delph (Getty Images)
Delph was supported by England manager Gareth Southgate in his decision to leave the camp before Tuesday's tense last-16 victory over Colombia on penalties.
"I've just experienced the most amazing 24 hours," Delph posted on Instagram.
"Watching my brothers come through a tough game both mentally and physically, I kicked every ball and felt every bit of emotion with the players, staff and fans.
"At 07:52 Me my wife and 2 beautiful daughters welcomed their baby sister into the world."
In contrast to Delph's decision, Sweden captain Andreas Granqvist has said he will definitely stay in Russia to play at the weekend even if his wife goes into labour.
"I'm planning to stay. My wife is a very strong person and we knew the situation before I came here," Granqvist told reporters after a 1-0 last-16 win over Switzerland.
"Nothing has happened so far... the baby doesn't want to come yet. We'll see what happens but I'll play the game for sure."