Morrison said claims were an "outrageous fabrication".
A lawyer for the Sunday Independent apologised to Morrison and his long-term partner Michelle Rocca in the High Court in Belfast on Wednesday.
The allegations were contained in extracts from an interview with Ms Lewis who appeared on Ulster Television's weekly chat programme, Kelly, last month.
A solicitor, reading out an agreed statement in settlement of Van Morrison's libel action, said the paper fully accepted that there was absolutely no truth in the allegations.
He said the paper's publishers, Independent Newspapers Ltd, had agreed not to repeat the allegations and to pay substantial damages to Van Morrison, together with his legal costs.
The Belfast-born singer, 56, who lives mainly in Dublin, recently recorded a top-selling album with Ms Lewis, a sister of legendary rock artist Jerry Lee Lewis.
After the UTV appearance, he issued a statement denying her claims.
"These outrageous allegations are a complete and utter fabrication," he said.
Morrison's solicitor said: "Hopefully this full retraction and apology will put an end to the publication of these totally false allegations in the press."
He said a libel writ had been served on UTV and a detailed statement of claim was being filed on Wednesday.
"We have also put on notice The Mail on Sunday, the Irish Star and the Daily and Sunday Mirror our intention to issue libel proceedings if we do not receive a satisfactory response and proposals forthwith.
"Mr Morrison will also be seeking substantial damages and we have instructions to issue libel proceedings if we do not receive an immediate and satisfactory response."