Pichot lost to incumbent Beaumont in the race to be chairman of the sport's governing body. AFP/File/Juan Gasparini
The 45-year-old former Argentina captain lost by just five votes (28-23) to incumbent Beaumont despite entering the race relatively late.
Pichot said he would also be resigning from all of his other administrative roles in the sport.
“As of today, I have decided not to continue as a UAR representative on the World Rugby Council, nor as a member of the RWC Board, nor as president of Americas Rugby,” Pichot said in a statement on the Argentina federation (UAR) website.
Pichot, who retired from the sport 11 years ago and had been World Rugby vice-chairman since 2016, hinted at trying for the top position again in the future.
“My proposal, which I fervently believe in, has not been chosen, and that is why I choose to step aside,” he said.
“I cannot conceive of occupying a place just for the sake of occupying it.
“Although I will now be exclusively dedicated to family and personal matters, this does not mean that I am moving away from rugby. I will always contribute, from the place that touches me.”
Former scrum-half Pichot helped make Sevens an Olympic sport and also elevated Argentina into the Rugby Championship and the Jaguares into Super Rugby.