The former Colorado Rapids coach will take over the npower League One side after Westley moved to Preston North End earlier this month.
Stevenage chairman Phil Wallace told the club's official website: "I understand this is a different approach but I believe it's a fresh, exciting appointment in the Stevenage way. We want young management who are motivated to succeed.
"Gary is young but he has already won silverware at the highest level possible in the US and now wants the opportunity to make his mark on the English game. I'm sure the experience of winning a championship in the US has added to his all round skill set and will serve him well.
"He has the values we seek at Stevenage and we look to the future with excitement as we continue to build the Club off the pitch as well as on it."
The 43-year-old will replace caretaker boss Mark Roberts at the helm.