Mackay and Iain Moody, Cardiff's former head of recruitment, exchanged texts and emails that were then made public by the club, leading to accusations of anti-semitism, sexism and homophobia.
As a result, Moody resigned as Crystal Palace's technical director and Mackay's widely expected appointment as manager at the London club was abandoned.
A statement from the FA said its policy was "to not bring charges in respect of private communications sent with a legitimate expectation of privacy".
(Reporting by Steve Tongue; Editing by John O'Brien)
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