In April, the High Court in London heard that Grant had settled his case against NGN because of the risk of a STG10 million ($A20 million) legal bill if his case went to trial.
British tabloid newspaper The Sun is published by News Group Newspapers. (AP PHOTO)
Grant, who is part of the Hacked Off campaign group, told BBC Radio 4: "Well, obviously, in light of these findings, we think that the CPS and the police should launch a new criminal investigation into this.
"And that was, as I say, the aim of Prince Harry's case, as I understand it, and certainly was my original aim in my case.
"And also, quite clearly now there should be Leveson Two. This is something that was repeatedly promised by the Labour Party in opposition to victims of press abuse over and over again, and now it suddenly seems to have disappeared from their priority list now that they're in government."
Tom Watson, former Labour deputy leader, also settled his claim this week as he was also offered a "full and unequivocal" apology and substantial damages for "unwarranted intrusion" into his private life during his time in government by the News Of The World.
Former deputy leader of UK Labour Tom Watson was also issued with an apology from NGN. (EPA PHOTO)
Speaking outside the court, Watson questioned whether the head of the Metropolitan Police, Mark Rowley, would take action when he receives a dossier of information linked to the case.
Rowley was asked on Friday if there will be a criminal investigation into NGN.
"There was a massive series of investigations done sort of a decade or so ago, finished more recently than that... many, many millions of pounds.
"Hundreds of officers have been involved for a long period of time. There were many prosecutions. Those investigations were closed.
"If they send us a bundle of material, we'll reflect on that and make our judgment.
"Much of the material in the civil litigation actually came from those investigations and was requested through legal processes by the litigants."
UK Culture Secretary Lisa Nandy confirmed that a second-stage Leveson inquiry has already been ruled out.
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