Steve Messham made a statement offering his "sincere and humble apologies" after lawyers acting for the peer threatened widespread legal action against those who named him on the internet or in the media.
Mr Messham said: "After seeing a picture in the past hour of the individual concerned, this (is) not the person I identified by photograph presented to me by the police in the early 1990s, who told me the man in the photograph was Lord McAlpine. I want to offer my sincere and humble apologies to him and his family."
Last night, the BBC also apologised "unreservedly for having broadcast the report".
Earlier, Lord McAlpine vehemently denied the "wholly false and seriously defamatory" claims against him. He has been the subject of speculation and innuendo since Mr Messham claimed last week he had been abused by a senior Conservative from the Thatcher era.
Representatives for the peer said solicitors were preparing writs with a view to taking legal action against "all media who have defamed Lord McAlpine's reputation and published defamatory statements".
In his statement, Lord McAlpine, who was born in London but has a strong Scottish heritage, said he had visited Wrexham "only once" with the trip made in the company of an agent from Conservative Central Office. They visited a relative of Lord McAlpine's and did not stay overnight, he said.
He said: "I have never been to the children's home in Wrexham, nor have I ever visited any children's home, reform school or any other institution of a similar nature. I did not sexually abuse Mr Messham or any other residents of the children's home in Wrexham."
Lord McAlpine said he had been defamed by "ill- or uninformed commentators" on the internet and "by innuendo" in the written and broadcast media. A "substantial number of people" would have "reasonably inferred" the allegations in the media had referred to him.
He said: "My name and the allegations are for all practical purposes linked and in the public domain and I cannot rewind the clock. I therefore have decided that in order to mitigate, if only to some small extent, the damage to my reputation I must publicly tackle these slurs and set the record straight."
Lord McAlpine's solicitor, Andrew Reid, said the peer had decided to speak out after presenter Philip Schofield tried to pass a list of alleged paedophiles to Prime Minister David Cameron live on ITV's This Morning. The names on the list were briefly caught on camera.
Mr Reid attacked BBC2's Newsnight programme, which last week broadcast Mr Messham's allegations without stating whom they referred to. He said: "They took what I think is the coward's way out. They ran the programme, trailed it, and then told everyone where to go and look for the name. They have done a very good job in severely damaging Lord McAlpine's reputation."
The BBC said last night in a statement: "On November 2, Newsnight broadcast a report that looked into criticism of the North Wales Abuse Tribunal.
"The report included an interview with Steve Messham, an abuse victim who said that a senior political figure of the time had abused him.
"We broadcast Mr Messham's claim but did not identify the individual concerned. Mr Messham has tonight made a statement that makes clear he wrongly identified his abuser and has apologised.
"We also apologise unreservedly for having broadcast this report."