Marler, who is not part of the England squad playing in Australia, posted his remarks after Dwyer criticised England's scrummaging during their surprise victory on Saturday.
Marler quickly deleted the comment.
"The RFU can confirm Joe Marler has been issued with a formal warning under RFU regulation 19.6.14, following his comments made on social media about former Wallaby coach Bob Dwyer on Tuesday 14 June," the RFU said in a statement.
"Marler is extremely apologetic for the comment that was made and deleted it within a few minutes.
"The warning will be retained on Marler's disciplinary record for a period of five years and may be relied upon in any future disciplinary proceedings."
Eddie Jones' side will seal England's first series win in Australia if they win the second test in Melbourne on Saturday.
(Reporting by Martyn Herman; Editing by Toby Davis)
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