The former Socialist MSP was awarded £200,000 damages, but was found guilty of committing perjury in 2010 in his evidence at the 2006 trial.
He was jailed for three years, serving just over one.
He has never received the £200,000.
The case was heard by Lady Paton, Lord Drummond Young and Lord McGhie.
For News Group, Roddy Dunlop QC said the application for an appeal to the Supreme Court was “a matter of extreme public importance”.
In May of this year, an appeal for the defamation verdict to be overturned was refused.
Mr Dunlop said: “The Court in its judgment of these matters had taken the wrong turn.” If the 2006 jury had known that Sheridan was an adulterer, hypocrite and perjurer, he said, they might not have found in his favour.
He added: “The verdict complained of could have been reversed by a jury which knew the truth. No court can close its eyes. This case simply warrants the attention of the Supreme Court.”
Appearing for Mr Sheridan, Donald Findlay QC said the 2006 jury was manifestly entitled to do what it did.
News Group, he said, was just seeking to avoid having to pay the damages.
The jury members, he said, were perfectly entitled to go ahead and decide on the evidence led that Sheridan had been defamed.
The verdict of a jury could only be overturned in exceptional circumstances, he added, and to overturn the verdict of a jury was one of the highest decisions one could make.
Mr Findlay described the attempt by News Group to appeal to the Supreme Court as “the last gasp of a dying man”.
After retiring to consider their decision, Lady Paton said that the Inner House could only grant permission for a case to go to the Supreme Court if the appeal raised a point of law of public importance, and they were not of that opinion. “As a result we refuse the application, but of course, it is always open to the applicants to go to London and plead directly to the Supreme Court,” she said.
After the hearing, Mr Sheridan said: “Yet again News Group had been in court attempting to pervert the course of justice. They are still and always will be this criminal enterprise.”