The 29-year-old centre is expected to be out of action for up to nine months following the procedure at BMI Ross Hall Hospital.
Richie had only just returned from another lay-off when he was injured playing for the Warriors against the Dragons earlier this month.
Via his Twitter feed Richie said: "I would like to thank the Warriors fans for the amazing ovation when stretchered off last week and the kind messages I've received since."
Richie added: "Sitting in hospital after successful Achilles surgery yesterday. Biggest worry at the moment is how skinny my calf is for wedding in July!"
Glasgow Warriors head coach Gregor Townsend said: "We're disappointed for Richie that he faces another spell on the sidelines, not long after he'd worked hard to get back to full fitness.
"The surgery was a success and we wish him all the best with his rehab."