Delaney, who is head of the Football Association of Ireland (FAI), said last month that he had no role or involvement in the OCI's handling of ticketing arrangements and would consider his position as the FAI's member of the OCI Executive Committee.
"Having given the matter careful consideration, I have made the decision to step down with immediate effect," Delaney said in a statement on the FAI's website (www.fai.ie).
"I am fully confident that in due course my position and non-involvement in these matters will be clarified and independently verified."
The OCI and Irish government launched separate independent probes into the circumstances surrounding the distribution of tickets at the 2016 Games and Delaney said he has fully engaged and co-operated with both investigations.
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Acting OCI president, Willie O’Brien, said in a statement that he accepted with regret Delaney's resignation.
Hickey, who was arrested in August at a luxury beachfront hotel and spent time in a maximum security prison, temporarily stood aside as head of the OCI and as Europe's top member of the International Olympic Committee (IOC) pending the outcome.
He has to remain in Brazil until further investigations into the charges are completed. No date has been set for his hearing.
Delaney, whom Hickey had tipped to succeed him before the ticket scandal plunged the OCI into crisis, said last month that Rio police had not contacted him despite an order being issued for the seizure of his passport.
(Editing by Ken Ferris)
