"I feel it is important that I acknowledge to you, our shareholders, that I made a mistake in becoming involved in an issue which I should have left to the business to deal with," Staley told the meeting in London.
"I have apologised to the board, and I would today like to apologise to you as well, for that error."
Barclays last month revealed that Staley, a 60-year-old US national, is being probed by regulators and will suffer a "very significant" pay cut over the incident, which comes as the bank struggles to repair a reputation battered by its role in the foreign exchange and Libor interest rate rigging scandals.
Staley requested that a team at the lender find who wrote the letter, viewing it as "an unfair personal attack on the senior employee".
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Barclays chairman John McFarlane acknowledged a lack of judgement by Staley but gave the chief executive his support to carry on in the top role.
"You know me, if I thought the chief executive should go, he would go. But I don't believe that is what should happen," McFarlane told the meeting.
Some 62.6 per cent of shareholders took part in the vote on whether to reappoint Staley as a director of Barclays, backing him almost unanimously.
Other directors were unanimously re-elected on turnout of around 72 per cent.