The 26-year-old made 138 appearances in four years for the Magpies, winning the Championship and playing in Europe in the process, and leaves with his head held high
The 26-year-old joined the Magpies on loan in 2009 before making the move permanent a year later.
Simpson made 19 Premier League outings for last season but found it difficult to secure a starting place ahead of Mathieu Debuchy and his now heading to Loftus Road in search for first-team football.
"The whole four years has just been class," Simpson told the club's official website. "I always say if anything's not gone right, then I've learnt from it and it's made me better. So I don't really have any regrets. Anything that's not gone right for me, I've learnt from and worked on and become better from it."
Simpson says he will cherish his time at St. James' Park, noting the experience of European football and winning the Championship.
The former Manchester United defender also notes his unbeaten run against rivals Sunderland as he prepares for life in England's second tier.
"Obviously winning the Championship the way we did is one of the highlights, getting back into the Premier League, but also just cementing my place in the team the season before last and finishing fifth.
"And there were one-off games - beating Sunderland 5-1, the comeback against Arsenal... I'll never forget those days and those games, but the whole four years have been top class.
"I don't think that 4-0 to 4-4 will ever happen again and to beat your rivals 5-1, that was my second Premier League game for Newcastle and to beat them the way we did that day, I know how much it means to the fans here. I can happily walk away and say I played four derbies: won two, drew two so I'm happy with that record.
"I loved playing pretty much every game when we finished fifth and got into Europe, and then playing in Europe as well - Benfica away, those kinds of games and experiences are invaluable and I loved every minute of it."