The malfunction, which National Air Traffic Services (Nats) said was caused by incoming flight data, saw more than a quarter of UK departures and inbound services cancelled on Monday (28 August).
Speaking to the Press Association, Lundgren said an incident on this scale "should not have happened" and "must not happen" again in the future.
"Passengers deserve to see a full independent review, which not only results in meaningful improvements to prevent an incident of this scale happening again but also considers a wide range of issues beyond this incident," he continued.
Nats is currently conducting its own investigation and will send a preliminary report to transport secretary Mark Harper early next week.
Earlier this week, Iata said the failure of the ATC systems is "yet another" example of why the passenger rights system "isn’t fit for purpose".
