Scottish Labour’s Westminster spokesman believes a new constitutional debate will “soon be upon us” and wants his party to “seize the moment”.
While attacking the prospect of a second independence referendum, Mr Murray told the Scottish Fabians Conference in Edinburgh that Scottish people want “strong devolution as part of the United Kingdom” while maintaining a strong relationship with Europe.
He said: “It is now clearer than ever that the First Minister intends to use the result of the Brexit referendum to continue to argue for a second independence referendum that most of Scotland doesn’t want and Scotland certainly does not need.
“Do we really want to divide ourselves again? As a nation, we are already deeply divided as a consequence of the independence referendum, and across the UK, the bonds that joined Scotland and England, Wales to England, and the north of England to the south of England, are at risk of fracturing.
“Our country has never been so divided. And we cannot meet division with division.
“We can only meet these challenges together, just as we have always done. But we have to recognise where something needs to be fixed.
“The Labour Party is not, and never will be, the party of the status quo. We are the reformers. And in this next period we must be the ones who present not just the old ideas for how Scotland can continue as part of the UK, but also how we do that with a new relationship with Europe.
“Brexit means that a new constitutional debate - not just one about independence, but about where power should lie - will soon be upon us.
“Let’s seize that moment and come up with a solution that meets what people across Scotland are asking for - strong devolution as part of the United Kingdom and maintaining a strong relationship with Europe.”
Mr Murray added: “We also must use this moment that Brexit has presented to ask how we can reform the UK, and in doing that no stone can be left unturned.
“We need to look again at the case for voting reform, and stop gerrymandering boundaries. And we must again restate our belief that the House of Lords should be replaced with an institution that meets the needs of a major democracy in the 21st century.
“Theresa May’s vision for the UK isn’t one that I think the majority of people will agree with. Our challenge now - at this turning point in our history - is to show that Labour - and the progressive left across Europe - can come up with solutions that speak to people’s lives.”