Hacked Off, the group campaigning for a free and accountable press, welcomed the "long overdue" rejection of the press' proposal for a Royal Charter.
The group's director, Brian Cathcart, said it was "regrettable" that further changes would be made to the cross-party charter and added that he would be watching closely to ensure there was no "dilution" of the Leveson recommendations.
Asked what would happen if the press decide to go it alone, he said: "If that happens we end up with another Press Complaints Commission, another failed non-regulator, we end up with more abuses and scandals.
"I'm afraid we'll all be back here again if that happens ... and in between there will have been more unfortunate victims of press abuses."
Hugh Grant has tried to counter what he said were portrayals of his Hacked Off campaign is a "smooth and sinister operation".
He told the BBC's Andrew Marr Show that it actually consists of "a few dandruffy professors sitting in a cheap office with a sightly insane chess champion ex-Lib Dem MP and a couple of threadbare lawyers... and me".
Support for a petition demanding that politicians implement the Leveson Report recommendations into press regulation has topped 100,000.
The petition urges David Cameron and Nick Clegg to, "implement, as soon as possible, the recommendations of the Leveson Report in full.
"Ignore pressure from media barons and introduce legally-backed regulation, independent of politicians and the press.
"Place tighter limits on how much of our media an individual is allowed to own, and
"Promote investigative journalism through effective public interest defences."
The campaign was organised by Hacked Off, the group campaigning for victims of phonehacking.
Hacked Off, the group campaigning for victims of phonehacking, has reached 20,000 signatures in its petition to support the implementation of recommendations provided by the Leveson Report.