Red Bull asked the FIA to review a 10-second penalty given to Hamilton after a crash that saw Verstappen smash into the barriers at high speed during the first lap of the British Grand Prix.
Verstappen was taken to hospital for checks after a 51G impact with the tyre barrier at Copse Corner and Red Bull argued the punishment handed to Hamilton - who went on to win the race - was not severe enough.
Red Bull boss Christian Horner accused the Brit of "dirty driving" at the time and Formula 1®'s governing body confirmed on Tuesday the team had submitted a petition for a review of the incident.
The FIA's International Sporting Code only permits requests for a review if "a significant and relevant new element is discovered which was unavailable to the parties seeking the review at the time of the decision concerned."
Red Bull supplied four pieces of evidence, but the stewards did not deem the information provided qualified as a "significant and relevant new element", so Hamilton's win at Silverstone stands.
Mercedes took aim at Red Bull after the verdict was revealed.
A statement from the team said: "In addition to bringing this incident to a close, we hope that this decision will mark the end of a concerted attempt by the senior management of Red Bull Racing to tarnish the good name and sporting integrity of Lewis Hamilton, including the documents submitted for their unsuccessful right of review.
"We now look forward to going racing this weekend and to continuing our hard-fought competition for the 2021 Formula One World Championship®."
Horner said Verstappen's crash with Hamilton cost Red Bull around $1.8million, an outlay that will have "massive ramifications" for the team.
Hamilton heads into this weekend's Hungarian Grand Prix just eight points behind Verstappen after triumphing on home soil for a record-extending eighth time.