One of David Cameron's more decent instincts is to protect his team from the wolves. All prime ministers detest the press pack forever at their door, trying to prove its bully-power is greater than any government. Like the Mounties, some editors boast they always get their man – or, in this case, their woman. Destroying a minister is a trophy hunt, this time with an added zest of revenge. Cameron circles his wagons round Maria Miller – but he's badly wrong on this.

The harm done to politics by the expenses scandal is felt by every MP in the blowback on the doorstep. Even the cleanest get the blame. Miller's behaviour confirms the worst people think of politicians. How a £1.2m London property housing her husband, children and parents could be called a "second home" defeats most reasonable people. All those "second" bedrooms strike a wicked contrast with the bedroom tax. If her MP colleagues cutting a £45,000 payback to £5,800 was astounding, her 32-second stroppy teenager non-apology took the biscuit. Cameron should have sacked her that day, not for his government's sake but to salvage a crumb of respect for the politician's trade.

But note how shrieking weekend headlines were all in the Tory press, led by the Mail on Sunday's "Sack her! 80% want PM to axe shamed minister". What's afoot? Their beef is against her role as minister responsible for Leveson press regulation, which they reject despite overwhelming backing from public and all parties. Britain's press, dominated by tax-shy oligarchs abusing their power for their own ends, has set itself against all who seek to rein them in.

Sadly, Miller has given them the perfect target, with her dumb spad warning the Telegraph against investigating Miller's expenses while she was considering press regulation. Look, they say, see what happens when government interferes with press freedom. But beneath that lurks their own threat: see what happens to ministers who mess with us.

It's hard to know who is most despised as politicians and the press vie for bottom place in public trust. But what absurdity, when one newspaper leader after another sanctimoniously thunders against MPs on the Commons standards committee for "marking their own homework". Yet when it comes to the press, they don't want anyone marking their homework at all. After the Milly Dowler case led to shutting the News of the World, Rupert Murdoch's "most humble day of my life" lasted no time at all. Nothing changed.

With all-party support, Miller legislated for a royal charter to set up an independent recognition panel to approve a press self-regulator, but that's rejected by most of the press. Instead, the press is winding up its discredited Press Complaints Commission and setting up the Independent Press Standards Organisation (Ipso) which will not seek recognition under the royal charter. This month it appoints a chair and a board.

The Guardian, the Financial Times and the Independent stay outside Ipso – and outside a rival organisation, Impress, set up as an alternative self-regulator that may seek royal charter recognition. (My stepson-in-law works for Impress, but check the archive for my strong support for proper regulation of our rogue press.) The royal charter confers protection on newspapers, but those who stay outside may land exemplary damages if sued. Any litigant can take a swing at a newspaper with no fear of paying lawyers' fees: win or lose, a paper not under royal charter protection will pay costs, even in failed cases.

Newspapers refusing royal charter recognition will challenge this law. It's a delight to hear those purveyors of disinformation on human rights and Europe – the Mail, Telegraph, Times and Sun – calling on the Human Rights Act and the European court of human rights to deliver them from this monstrous intrusion on their freedom. Note how none of these papers – not one – backed the Guardian when we came under direct government threat over Edward Snowden's exposure of genuine Big Brother surveillance.

Whose interpretation of a "free press" do you trust? Last month a letter organised by the Hacked Off campaign called for the press to "accept a form of self-regulation that is independently audited" under the royal charter. Among its signatories were Helena Kennedy QC, John Pilger, Nick Davies, Will Hutton, Judge Stephen Sedley, Michael Mansfield QC, Peter Tatchell and Professor Conor Gearty – hardly lackeys of the oppressive state. Speaking truth to power is an essential freedom, but that's not the same as the right for media moguls to disseminate false information in pursuit of their private interests. The warped ownership of most of Britain's press is an unaccountable private power. Murdoch or his men met Cameron 26 times after 2010, as he tried to acquire BSkyB.

The PCC code drawn up by editors was always exemplary. Look at its first article: "The press must take care not to publish inaccurate, misleading or distorted information, including pictures." All that's needed is for the press to abide by its own code, yet it's broken every day. The nasty nexus of interests revealed in the Miller case shows why public opinion is right: neither MPs nor the press are fit to regulate themselves.