Cardinal Keith O'Brien sent the apology to Peter Campbell from Greenock after alleging he was responsible for the nuisance calls earlier this year.
A police investigation found another man was thought to have made the calls to his Edinburgh home.
The church said Mr Campbell had made unsolicited calls to him in the past.
"The cardinal alerted the police and also wrote to Peter Campbell inquiring as to whether he was responsible and asking him to desist if he was.
"He took this step because Mr Campbell has a history of making frequent unsolicited calls to the cardinal."
"It now seems that another caller made the abusive calls in question - which are currently a police matter."
Detectives were called in by Cardinal O'Brien following a spate of abusive calls in February.
'False allegations'
Investigators succeeded in tracking down the caller, who is said to have a history of drug abuse.
Prosecutors are now considering whether to pursue a case against him.
A police spokeswoman said: "Lothian and Borders Police have reported the matter to the procurator fiscal."
In a letter, Cardinal O'Brien told Mr Campbell: "I apologise to you most sincerely for having made allegations against yourself as being the perpetrator of the abusive telephone calls to my home.
"I apologise for any hurt which you or your family may have sustained through these false allegations.
"I now realise that it was not you who was responsible in any way and I am glad that the perpetrator has been found."+++++++++++++++++++++++++++++++++++++++++++++++++++
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(Source: BBC)