Just when you thought it was safe to go back on to Twitter, Trump strikes again.

His allegation that his predecessor Barack Obama ordered a wiretap of his phone during the election campaign is the most bizarre moment yet in a presidency that, barely a month in, is already littered with them.

Any lingering hope among mainstream Republicans that Trump in the White House would be different from Trump on the stump has surely been exploded by his latest outburst. This is despite an appearance before Congress which saw him widely praised for behaving like a normal president; he simply can't help himself.

Neither Trump nor his team has provided any evidence to back up his claim, which if true would see both Obama and the guilty parties from the FBI behind bars. And the call by FBI director James Comey for the Justice Department to publicly reject Trump's allegation places the president in a potentially very awkward position. He has apparently libelled Obama and traduced the FBI, reducing his already poisonous relationship to the intelligence services to a new low. Neither can afford to let the allegations go unanswered, but Trump is not known for gracious apologies when he has been shown to be wrong.

So this really matters. It's gone beyond the random abuse of people Trump thinks have done him or his family down. It indicates an indifference to the importance of evidence that's deeply worrying in a role where being able to weigh the evidence is of fundamental importance. He heard an unfounded allegation on a right-wing news site and swallowed it hook, line and sinker, without asking even the most basic questions about it (like, 'Really, how likely is this to be true?').

Trump's character flaws are very well known. They didn't stop large numbers of Americans, including very large numbers of evangelicals, voting for him. They thought he'd be competent, or that he'd surround himself with competent advisers who would restrain his worst excesses. But what's just happened is so serious that no one can honestly hold that view any more. The legal consequences of what he's done aside, his attack on Obama and the FBI is so bizarrely inappropriate that his unfitness for office can no longer be in doubt.

The trouble is that many evangelicals have backed themselves into a corner when it comes to their support for him. They believe God ordained the result of this election. Trump, they think, is God's man for the hour. Some of them think this because they have a warped understanding of the sovereignty of God and the application of Romans 13:1, 'Let everyone be subject to the governing authorities, for there is no authority except that which God has established. The authorities that exist have been established by God.' Some of them continue to back him because they like what he's said he'll do – building a wall, increasing spending on the military, dismantling Obamacare.

The first reason is manifestly inadequate (ironically, a literal reading of that text would have forbidden the rebellion that established the US in the first place). It's a licence for tyranny and throws democracy itself into question. No protests, no petitions, no nothing: God ordained your ruler, and questioning him is questioning God. The second requires just the sort of competence and attention to detail which every day of Trump's presidency shows him to be lacking.

So when will the scales fall from evangelical eyes? When will high-profile backers like Franklin Graham start calling him out when he does things that are obviously wrong, as well as praising him for the things they think he's getting right? Because at the moment, US evangelicals are in the horrible position of identifying themselves with a man America is realising cannot do the job it elected him to do. When he falls – and that's looking increasingly likely – the consequences for those who've hitched their wagon to him will be grim.

Follow Mark Woods on Twitter: @RevMarkWoods