Elliot, a high-profile anti-racism campaigner, acknowledged sending a text message to former Charlton defender Richard Rufus in which he used a "term which is widely known as being derogatory to my own community." Both men are black.
The FA said the "discriminatory abusive comments" made Elliott's position untenable.
As well as leaving the FA's judicial panel, Elliot is relinquishing his position on UEFA's football committee, and a key role with English football's anti-racism body, Kick It Out.
"I am saddened by this turn of events and it is with regret that we accept Paul's resignation," FA chairman David Bernstein said. "However, the use of discriminatory language is unacceptable regardless of its context and in effect has made Paul's position untenable."
Elliott said he regretted using the offensive comment in the mobile communication.
"It is inappropriate and not part of my everyday vocabulary," Elliott said in a statement released by Kick It Out. "As an advocate of high-standards of public behaviour, and integrity in public life, I know the use of this word sends out mixed messages and contradicts my position as a Kick It Out trustee.
"I will continue to be active in other projects in what I believe to be a true and just cause."
Earlier this month, Elliott became the first black footballer to receive the Commander of the Order of the British Empire award at Buckingham Palace for services to equality and diversity in football.