The 60-year-old had been the favourite to take over from Sam Allardyce and held talks with the Tyneside club on Friday night.
However, after considering the move overnight, Redknapp arrived at Pompey's Eastleigh training ground on Saturday morning - the team are due to depart for the North-East to play Sunderland on Sunday - and revealed he was staying with the club.
Redknapp told Sky Sports News: "I'm happy - this is a club I feel comfortable at. I have a fantastic relationship with the fans, a fantastic relationship with the owner and with Peter (Storrie, chief executive).
"The players I've brought to the club, I've sold Portsmouth to them and to walk away having brought them here, when the club is doing so well, wouldn't be right."
But the 60-year-old concedes the offer from Newcastle was both flattering and tempting.
"I had a great offer from them and to be offered the chance at a fantastic club like Newcastle was a terrific opportunity for me but at the end of it I couldn't do it," told Sky Sports News.
"I had no intention of leaving here. The job now is to take this club forward. To walk away is not the right thing to do."