The leaked document contained a second-hand account of the conversation and claimed that Ms Sturgeon had said she would prefer David Cameron as prime minister compared to Ed Miliband.
Both Ms Sturgeon and the French ambassador denied the version of events presented in the document.
“I had not seen the document before it was published in the Daily Telegraph, however I was aware of its content and agreed that my special adviser should make it public,” Mr Carmichael said in a statement.
“I should not have agreed this. It was an error of judgement which I regret. I accept full responsibility for the publication of the document.
“I have written today to the First Minister and to the French Ambassador to apologise to them both; had I still been a Government Minister I would have considered this to be a matter that required my resignation.”
"I accept his apology, but the real issue is that he should be apologising to the people of Orkney and Shetland, because he clearly contested the election on false pretences."