Juan Miguel Villar Mir, who finished third in last July's poll of club members, said he wanted to avoid damaging Real Madrid, news agency Europa Press reported.
Allegations of vote-rigging led 10,511 mailed ballot papers to be provisionally disallowed days before the election, reports AP.
Villar Mir received 6,702 of votes cast in person 1,396 fewer than Calderon but also claimed at least 4,700 of the postal votes.
Another beaten candidate, Arturo Baldasano, has yet to announce whether he will appeal the ruling. Baldasano came last out of five candidates.
Almost 28,000 members voted in the hope of electing a new administration which would end Madrid's three-year drought without a trophy, its worst streak since the early 1950s.