Aside from serving on FIFA's top body, the Kuwaiti heads the influential Association of National Olympic Committees, the Olympic Council of Asia and other senior sport administration posts, including with the Asian Football Confederation (AFC).
Sheikh Ahmad has been named as a co-conspirator of disgraced Guam football chief Richard Lai, who pleaded guilty in New York this week to receiving nearly $1 million in bribes from football officials wanting his help to influence FIFA.
The Kuwaiti denied all wrongdoing on Saturday before deciding to step back from his FIFA post on Sunday.
"I do not want these allegations to create divisions or distract attention from the upcoming AFC and FIFA Congresses," he said in an emailed statement.
"Therefore, after careful consideration, I have decided it is in the best interests of FIFA and the AFC for me to withdraw my candidacy for the FIFA Council and resign from my current football positions," he added.
"I will continue to support the family of football once these allegations have been disproved."
- Resignation 'best' for FIFA' -
FIFA president Gianni Infantino, in a statement, said he had "taken note" of Sheikh Ahmad's resignation.
"I want to thank him for taking this decision which certainly was not easy to take but is in the best interest for FIFA."
Sheikh Ahmad, a former Kuwaiti oil minister, has been linked to bribes accepted by Lai in exchange for backing an Asian candidate in FIFA's 2011 presidential election and help in gaining control of the AFC.
Documents filed with a federal court in Brooklyn identify Lai's "co-conspirator #2" as "a high-ranking official of FIFA, the Kuwait Football Association, and the Olympic Council of Asia".
Sheikh Ahmad is not mentioned directly, although the documents filed on April 27 say all of Lai's alleged co-conspirators "are known to the United States Attorney".
Co-conspirator #2 is tied to multiple payments aimed at influencing football elections, including a November 2009 transaction worth $200,000.
Lai, Pacific island Guam's football chief since 2001, has been suspended by both FIFA and the AFC after he admitted accepting bribes between 2009 and 2014.