The MP for Edinburgh South was speaking on BBC Scotland’s The Sunday Show programme.
He told host Martin Geissler that “credibility, integrity and honesty” are something to uphold in parliamentary positions, adding society would be “degraded” if excuses are found “for a prime minister who has lied to the electorate, laughed at the electorate, made laws and broke laws”.
He said: “We’re going through probably the biggest cost-of-living crisis in the history of this country, where many are discovering whether they can pay their bills, whether or not they can make a decision about heating or eating, and we seem to be consumed by a Prime Minister who’s completely unfit for office.
“He’s not just been doing this as Prime Minister. Every career he’s had, he’s either been fired from or asked to leave because his integrity has been under question.
“I just don’t think you should have that at any position in parliament, let alone prime minister.”
Mr Geissler asked the MP why the Labour Party is “struggling” to increase support, telling him they should be “absolutely demolishing” the Conservatives in the polls following various scandals.
Opinion polls, he told Mr Murray, are putting his party in single-figure leads.
Mr Murray said: “Polls are polls. We have a long way back from the 2019 election, let’s not forget this government has an 80-seat majority.
We need to persuade the electorate that we're a government-in-waiting
“We’re making progress in terms of setting the agenda on how we deal with the cost-of-living crisis – we have a plan for that which the Government don’t.
“We have a plan for how we would like to take the economy forward, whereas the Government don’t. We have a plan for the green agenda, which the Government don’t.”
He added: “We need to persuade the electorate that we’re a government-in-waiting.
“The opinion polls have been fluctuating quite a lot – in fact, had we had this discussion just a few weeks ago, we’d have been in double-digit leads in the polls.”
Mr Murray said implementing a written constitution could provide more mechanisms to hold the Prime Minister to account rather than him being “judge and jury on himself”.
“At this moment in time, our entire democracy is being undermined by a Prime Minister who is dishonest,” he said.
“When you look at that in the round, what the Prime Minister’s trying to do is to tar everyone with the same brush and undermine democracy in order to get away with his own failings. That’s completely unacceptable.”
The Labour MP also defended his party leader, Sir Keir Starmer, after Mr Geissler suggested he was “too dull”.
Mr Murray said the Labour leader’s key attributes are “integrity, credibility, intelligence and honesty”.
“That’s what we need in a prime minister,” he replied.
“I think Keir Starmer would be a fantastic prime minister. He certainly wouldn’t embarrass the country on the international stage.
“He certainly wouldn’t be focused on things that the country aren’t focused on. He’d have a laser-like focus on the cost-of-living crisis.”