In a pitch for the deputy job, the former shadow Scotland minister said the buck would stop with him on tackling complaints of anti-Jewish hate and racism, sexual harassment and bullying.
Mr Murray, a long-time critic of Jeremy Corbyn, also called for an end to the vilification of dissenting voices and said Labour would suffer if the next leader surrounds themselves solely with allies.
In an interview with The Independent, the Edinburgh South MP said: “We need to rid the Labour Party of the cancer of antisemitism but it is much wider than that.
“It’s the bullying, it’s the intimidation, it’s the lack of tolerance, it’s the vilification – all of that stuff. I will take personal responsibility for that.”
He took a swipe at Mr Corbyn over the perceived failure to handle complaints of antisemitism, which prompted an outcry amongst Jewish groups and an unprecedented intervention by Chief Rabbi Ephraim Mirvis.
“The lack of – or perceived lack of – dealing with it has made it worse,” Mr Murray said. “If the Labour Party can’t sort itself out, how can the people trust us to sort the country out?”
The straight-talking Scot made a last-minute decision to enter the deputy race but he has already cleared the first hurdle by winning endorsements from the required 22 MPs.
Growing up on a housing estate in a deprived part of south Edinburgh, Mr Murray never saw himself entering politics.
Official tallies put him in a distant second place behind shadow education secretary Angela Rayner, who is seen as the candidate to beat after taking an early lead in the five-way race.
Shadow cabinet ministers Dawn Butler and Richard Burgon are also closing in on the minimum threshold, alongside shadow sports minister Rosena Allin-Khan.
Mr Murray may struggle to beat Ms Rayner, but he describes himself as a “winner and a fighter”, citing his record of holding off the SNP surge that wiped out his fellow Scottish Labour MPs in 2015 and again in December.
As the party’s only Scottish MP, Mr Murray said he had track record of beating the odds and his appointment would send a message on Labour’s intention to reach out into its former heartlands, in both the north of England and Scotland.
Mr Murray said: “Part of my pitch is that Scotland’s voice has to be heard at the top of the UK Labour Party. Not just because it’s Scotland… it’s a much wider issue than that.
“It sends out the message to the whole of the UK and every single region and nation of the UK, that the UK Labour Party is listening and taking them seriously.”
Mr Murray said he wanted to overhaul party structures to hand power back to the regions, to allow people in Lincolnshire, Cornwall, Cumbria and elsewhere to take the fight back to the Tories.
Labour’s sluggishness to take a position of Brexit and Scottish independence were at the heart of its electoral drubbing, he said, as well as its “wish list” manifesto that the public did not believe.
He said: “When you stand in the middle of the road on two big constitutional issues, you don’t win. You get knocked over.”
A prominent advocate for a Final Say referendum, Mr Murray said Labour must now fix its firepower on preventing a no-deal Brexit at the end of the year.
When pressed, he would not rule out campaigning to rejoin the EU but only if the public support was there.
He said: “If the future looks terrible then we will have to do something to save the country. That might be renegotiating a much better and closer relationship with the EU in the future, it may be that we have a debate about whether or not we want to be a member.”
Labour must be clear that it opposes Scottish independence, he said, as he condemned leadership hopeful Clive Lewis’s suggestion that the party would back another referendum.
Mr Murray said: “If leadership [contenders] want to open their mouths and let their stomachs rumble about big constitutional issues they genuinely know nothing about – because they told me they know nothing about it – the Labour membership can decide whether they want them to be leader of the Labour party.”
In a plea to leadership candidates, he said Labour must move on from the Corbyn years or risk another election disaster.
He said: “The next leader of the Labour Party has one of two choices – to turn the Labour Party into a credible alternative government, or to keep the Labour Party as a party that will just be in perpetual opposition.
“Continuity means we are ignoring the public and if we ignore the public we will get the same result again.”
The new leader and deputy leader of the Labour Party will be announced on 4 April.