He made the plea after Labour, SNP and Liberal Democrat MPs united with rebel Conservatives to inflict defeat on Prime Minister Theresa May in a key vote on the European Union (Withdrawal) Bill.
In light of that, Cable urged SNP MPs to back a Liberal Democrat amendment to the Bill which would provide for a referendum on the exit deal once it is agreed, giving the public the chance to vote for the UK to remain in the EU if they do not like the final agreement.
The Liberal Democrat leader said: “There is growing evidence that the British public wants a say on any final deal with the EU, including the option of an exit from Brexit.
“Moreover, the Scottish people voted to remain in the EU, so their elected representatives in Parliament really must push for a popular vote.
“Which is why the SNP’s Westminster party must show some backbone and support our amendment.
“Nicola Sturgeon herself has said a public vote would be ‘irresistible’ in the event of a ‘no deal’ Brexit that would see the UK crash out of the EU and that the people have a ‘right to look at the outcome’ if there is a deal.
“But she must go further and back our call for a referendum on any Brexit deal.
“The Conservatives have botched Brexit so far and the last thing the SNP should be doing is allowing the Conservative right to secure an economically damaging hard Brexit.”
An SNP spokesman said: “We could end up crashing out with no deal, or with a very bad deal, and, in those circumstances, for the UK Government simply to say to people they have to accept the outcome no matter how bad it is, may become very difficult for them to sustain.”