Lord Justice Leveson yesterday published his report following an eight-month inquiry into press ethics and standards following the phone hacking scandal at the News of the World.
[Left: McCanns / Right: Cameron]
It recommended a continuation of the system of press self-regulation, but via a new independent body underpinned by legislation.
The father of Madeline McCann - the young girl who went missing in Portugal in 2007 - said that he would have liked the report to have gone further in terms of regulation.
"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," Gerry McCann told reporters.
"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."
Lord Justice Leveson's 2,000-page report found that press behaviour had been "outrageous" and "wreaked havoc with the lives of innocent people".
He said that the press had failed to regulate itself in the past, but gave the industry another chance to create a tough new watchdog, backed by legislation to ensure that it was effective.
Mr McCann's family was the subject of "unbelievably damaging" newspaper reports, including stories that suggested he and his wife, Kate McCann, had killed their daughter Madeline.
Speaking to BBC Radio 4's Today programme, he called on Cameron to implement the full recommendations of the report.
"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," he said. "Without statutory underpinning, this system will not work."
[Left: Leveson / Right: Maria Miller]
Speaking in the Commons yesterday, Cameron said he broadly welcomed the Leveson recommendations, but had "serious concerns and misgivings" over bringing in a law to underpin the new regulator.
Most of the newspaper frontpages have supported the prime minister over this stance, although it has exposed divisions with his deputy Nick Clegg, who wants the new law to be introduced. Labour leader Ed Miliband has also backed the legislation.
Cross-party talks started last night and will continue next week, after which the Department for Culture, Media and Sport will start drawing up a draft bill for implementing the Leveson recommendations. This could be ready in a fortnight.
Culture Secretary Maria Miller said that "the gauntlet has been thrown down" to newspapers to create a system of tough self-regulation.
But she also told BBC Breakfast that Conservative ministers feel that legislation is not needed to "achieve the end of objectives".
She said that they are concerned that a 'press law' "would actually give the opportunity in the future to bring into question the ability of Parliament to stay out of the issue of free press and difficult for Parliament to not have a statutory framework on which they could hang further bits of legislation".
Miller added: "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."