The 84-year-old Paraguayan, who had heart surgery for the fourth time in December, has led Conmebol since 1986 and been on the executive committee of world body FIFA since 1998.
"I have invited you to this meeting to share with you my retirement from ... FIFA," Leoz told reporters before adding that he had advised president Sepp Blatter of his decision.
"I'm announcing my retirement from all FIFA committees because it's important for my life. A very important committee is the organising committee for the 2014 World Cup but important reasons prevent me from staying in that position.
"I still have good mental health but physically I'm often impeded," added Leoz who was accused in a BBC Panorama programme of receiving money from ISL, FIFA's former marketing partner.
He denies any wrongdoing.
The results of a FIFA investigation into allegations of bribery involving ISL were due in March but could now be presented before Congress in May.
Leoz said he quits with a clear conscience.
"I feel very happy because I'm retiring with the tranquility and knowledge of having done a sincere, honest job," he added.
Soccer's ruling body issued a brief statement following his decision.
"FIFA has taken note of the formal resignation of Nicolas Leoz as a member of the FIFA executive committee and as president of CONMEBOL for health and personal reasons," it said.
"In accordance with FIFA statutes CONMEBOL will now have to decide immediately on the replacement of Nicolas Leoz as one of its representatives on the FIFA executive committee."
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- FIFA executive committee
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