Rangers boss Harry Redknapp has agreed a three-year deal with Simpson, who is a free agent after his Newcastle contract expired, as he begins rebuilding his squad for the new Championship season.
The 26-year-old told Rangers' official website: "I want to play week in, week out. That was the main thing for me. Harry wants me playing and he wants me involved.
"He's a top manager, and as soon as I spoke to him I knew this was the right move for me.
"I've always admired him. He is a great man in the game and I have a lot of respect for him.
"From what I have heard from Harry and the staff I have spoken to, along with some of the players here who I know well, they are very determined to put things behind them and start afresh and get back into the Premier League where the club belongs.
"I want to play my part in that and do everything I can for the club and the fans. I really think we can go straight back up."
Simpson began his career with Manchester United and made 137 appearances in four seasons with Newcastle.
Redknapp added: "He's the type we want at the club. He's a good professional, a good character and a good player who knows his job in the side.
"He'll be a really fine acquisition for us. It's a fantastic piece of business for the club on a free transfer."