"I am not defending myself, what I did was wrong," the Briton said on his Twitter page (@louissmith1989). "I want to say sorry for the deep offence I have caused and to my family who have also been affected by my thoughtless actions.
"I have learnt a valuable life lesson and I wholeheartedly apologise."
The Sun newspaper had published a clip from the video, taken at a hotel wedding reception, showing the two gymnasts pretending to pray while shouting "Allahu Akbar" (God is greatest).
Smith is also shown with a drink in his hand.
British Gymnastics said in a statement that it "does not condone the mocking of any faith or religion and is appalled by such behaviours".
It said any members breaking the organisation's code of conduct risked suspension or expulsion and it would be investigating the reported behaviour.
Smith, 27, won pommel horse silver at both the London 2012 and 2016 Rio de Janeiro Olympics. He also won an individual bronze at Beijing in 2008 and a team bronze in London.
He is no stranger to controversy, however, and was warned in April about his conduct after a Twitter outburst questioning the judging of the national championships where he lost out to younger team mate Max Whitlock.
Whitlock won two gold medals in Rio. Carson retired from gymnastics last year due to injury.
(Reporting by Alan Baldwin, editing by Ed Osmond)
