Mr Fox said it would be impossible to argue that the SNP should not call a second referendum on Scotland's place in the UK if campaigners to say in the EU were given a chance to reverse the decision on Brexit.
His comments came as First Minister Nicola Sturgeon confirmed she would only reveal her thinking on a second independence referendum in the new year.
"How can we tell Nicola Sturgeon and the SNP that they can’t have another independence referendum because they didn’t like the result?” he warned.
The leading Brexit supporter added that "if there is a referendum... people like me will immediately start asking that it’s best of three”.
“I’ll set out my views in the new year once we’ve got through this period of turmoil on Brexit," the First Minister confirmed on Sky's Ridge on Sunday programme.