The 38-year-old was one of several players to leave The Hawthorns this summer following the Baggies' relegation from the Premier League and has now linked up with Steven Gerrard's Gers on a one-year deal.
"The opportunity to come and play at this famous club is amazing," McAuley told the Rangers website. "My family support the club, myself as well. The street I was born on, the kerbs were painted red, white and blue.
"There's a young group of players here. I want to help develop them and also challenge them to play in the team. Speaking to the manager in the last week or so, that's something I can bring to the project."
The Northern Ireland international, who has also previously played for Leicester City and Ipswich Town, will link up with his new side for the first time after the international break.