Welbeck was replaced by Jack Wilshere after 24 minutes of Monday's 2-2 Premier League draw with Manchester City at the Etihad Stadium.
Specialists determined surgery was necessary "after scans and tests revealed significant cartilage damage".
Arsenal expect a full recovery to take "approximately nine months", meaning he will be unavailable for selection for the European Championship in France next month.
Welbeck only made his return from nine months out with a similar injury to his left knee in February, and now faces another lengthy spell on the sidelines.
An Arsenal statement read: "Danny Welbeck has had surgery to his right knee. Specialists determined surgery was required after scans and tests revealed significant cartilage damage.
"Danny will miss the Euro 2016 championship in the summer and his full recovery is expected to take approximately nine months.
"Everyone at Arsenal will be working hard with Danny throughout his recovery and look forward to seeing him back on the pitch as soon as possible."
Despite his optimism, the news will come as a major blow for England manager Roy Hodgson. With Daniel Sturridge lacking regular football following a series of niggling injuries since the turn of the year, the former Liverpool boss could look to the likes of Andy Carroll and Jermain Defoe to provide depth in attack.
Following an impressive breakthrough season at Manchester United, Marcus Rashford's chances of earning a surprise first senior cap have also improved.
