Media and Sport will begin the process of drawing up a draft bill implementing the Leveson recommendations.
It is thought the draft legislation may be ready in a fortnight, the BBC reported.
Miller said: "the gauntlet has been thrown down" to newspapers to outline how they would set up tough self-regulation instead.
But Gerry McCann, the father of missing Madeleine McCann, said the Leveson report has not "gone far enough".
She went on: "At this point what we should be focusing in on is the fact that the gauntlet has been thrown down to the industry.
"The press industry need to be coming back with their response to the Leveson report. Their response to how they're going to put in place a self-regulatory body that adheres to the Leveson principles and that is what I want to see moving forward swiftly."
Many of Friday's newspapers have praised Cameron's opposition to law-backed regulation.
But the father of Madeline McCann - the young girl who went missing in Portugal in 2007 - said he would have liked the report to have gone further.
"Although we broadly welcome Lord (Justice) Leveson's report, and it has many merits, for me, personally, I don't think the report has actually gone far enough," said Gerry McCann.
He said: "I would have liked to have seen a properly independent regulation of the press, whereas I think he has given the press another opportunity of self-regulation".
McCann, who was the subject of what he called "unbelievably damaging" newspaper reports told BBC Radio 4's Today programme: "Full implementation of Lord (Justice) Leveson's report is the minimum acceptable compromise for me and, I think, for many other victims who have suffered at the hands of the press. Without statutory underpinning, this system will not work".