Northern Ireland could be handed new powers as a result of Brexit, Michael Gove has said.
The Tory leadership candidate said leaving the European Union offered the opportunity to "renew and reboot" the Union as he made his pitch to succeed David Cameron as Prime Minister.
Mr Gove said control over policy areas such as agriculture and fishing could flow to the Scottish Parliament, as well as the Welsh and Northern Ireland assemblies.
His pitch to develop a "fairly-funded, flexible and robust" Union was seized on by the SNP, who accused him of plotting to axe the Barnett Formula, the Treasury mechanism for distributing central funds between the nations of the UK.
Mr Gove, who was born in Edinburgh and brought up in Aberdeen, was speaking at a campaign launch a day after the declaration of his candidacy forced Boris Johnson out of the leadership race.
He said: "This referendum has led to questions about how we stay together in one United Kingdom - and for me, in every sense, this is about family.
"In a family you listen, you treat each other with respect, you make things better.
"That is what I will do. Treating Scotland, Wales, Northern Ireland with respect. Working to make things better. The vote to leave the European Union gives us the chance to renew and reboot the Union," he added.
"We are taking back control of policy areas like agriculture and fishing that are vital to the economies of Scotland, Wales and Northern Ireland, and the Scottish Parliament and devolved assemblies can enjoy new powers in these and other areas.
"I think we need to explore how we can develop a fairly-funded, flexible and robust Union for our new circumstances - and I will work across political divides, with respect, to build that new Union."
Mr Gove said the vote in Scotland, where 62% backed Remain, raised "profound" questions and vowed to listen to Scottish public opinion.
Commenting on the efforts of First Minister Nicola Sturgeon (left) to secure Scotland's place in the EU, he said: "I don't want to take any precipitous steps and the First Minister has an absolute constitutional right to do as she thinks is appropriate in these circumstances.
"If she wants a Prime Minister who understands and believes in Scotland - and indeed somebody who has got personal friends in the SNP, who include SNP MPs, people I've worked with for 20 years - then I can do that. I can do it because the one thing I will want to do is make the United Kingdom work and I will treat with respect those people who've got a mandate in Scotland."
Asked whether he would block a second referendum on Scottish independence if successful in his bid for the leadership, he responded: "I don't think we're going to have a second independence referendum."
SNP MSP Mike Russell, convener of Holyrood's finance committee, said: "People across Scotland - in every single part of our country - voted to remain in the EU and we are determined to make sure that clear democratic expression is recognised.
"It's absolutely outrageous that a prospective Prime Minister is now using a Leave vote to imply that Scotland's budget could be slashed just months after the Tories agreed a new financial settlement for Scotland."
He called on all leadership candidates, including Mr Gove's main rival, Home Secretary Theresa May, to rule out any cuts to Scotland's budget.