The Brentford player fell over advertising hoardings during celebrations after the 3-2 victory at the Estadio Azteca, which took them through to the quarter-finals.
The 36-year-old was taken to hospital, having left the field on a stretcher.
He is now set to miss the remainder of the World Cup but is expected to stay with the team at their Kansas City base.
"Surgery done. Let's get ready for the big one on Saturday," Henderson posted on Instagram.
"Thank you to all the staff that looked after me at Kansas City Orthopaedic Institute, especially the three surgeons who carried out the operation."
England face Erling Haaland's Norway in the last eight of the World Cup in Miami on Saturday.