Hickey, 71, was arrested at his Rio hotel after being accused of ticket scalping, ambush marketing and conspiracy.
He was taken away from the hotel in an ambulance amid concerns for his health.
The OCI statement said he had stepped down temporarily from his role with the Irish body as well as an International Olympic Committee member, president of the European Olympic Committees and vice-president of the Association of National Olympic Committees.
"In light of this morning's developments and his ill health, Mr Hickey has taken the decision to step aside temporarily as president of the OCI and all other Olympic functions (IOC member in Ireland, European Olympic president, ANOC vice-president) until this matter is fully resolved.
"Mr Hickey will of course continue to co-operate and assist with all ongoing inquiries."
Prosecutors have estimated the profits from the alleged black market sales at 3 million dollars.
"We will be fully cooperating with any police investigation," IOC spokesman Mark Adams told a press conference.
"Mr Hickey is entitled for the world and everyone to believe he is innocent until proven guilty," he added. "Let's not jump the gun on this one."
Brazilian police have launched a widening investigation after seizing more than 1,000 premier tickets for Games events, including the opening ceremony.
Tickets with a face value of about 1,000 dollars were sold for more than 8,000. Some of the tickets had the Olympic Council of Ireland name on them.
The OCI would only say that it was "seeking total clarity on the situation" before commenting.
Kevin James Mallon, a director of THG Sports, was arrested in Rio on August 5 in possession of the tickets. THG Sports was licensed to resell London 2012 and Sochi 2014 Olympic tickets. Hickey's son Stephen used to work at THG.
Brazilian prosecutors issued warrants for Hickey and other figures including Marcus Evans, a British hospitality businessman and owner of English Championship football club Ipswich.
Hickey has been a member of the IOC executive board since 2012. As head of the European body he has been instrumental in setting up the European Games which was first held in Baku, Azerbaijan, last year.