The Football Association heard the claim for wrongful dismissal but decided Atkinson had made the correct call.
Vermaelen, 24, will now miss Saturday's trip to Birmingham.
A statement released by the FA read: "At a Regulatory Commission hearing today, a claim for wrongful dismissal from Arsenal defender Thomas Vermaelen was dismissed.
"Vermaelen was shown a red card for denying an obvious goalscoring opportunity during Arsenal's match against West Ham United on Saturday 20 March.
"He will now serve a one match suspension with immediate effect."
Arsenal manager Arsene Wenger must now decide who will replace Vermaelen against Alex McLeish's side.
William Gallas continues to be sidelined by an ongoing calf strain with Sol Campbell deputising, so Wenger may be forced to turn to veteran Mikael Silvestre to form a partnership that will have a combined age of 67.
Vermaelen has been a key presence at the Emirates this season, scoring eight goals in 41 appearances.
The 24-year-old, signed for £10million from Ajax last summer, has proved a brilliant buy as Arsenal chase silverware on two fronts.
The Gunners sit two points behind leaders Manchester United with seven matches remaining and also face a Champions League quarter-final against Barcelona.
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