To what extent is Brexit legitimate?

The Term Brexit relates to the United Kingdom leaving the European Union. In June 2016, David Camerons government launched a national referendum asking the general public if they were in favour of the single market union. The referendum gave a definitive leave vote of 52%, however, is that actually legitimate? With a turnout of 72%, it is only 35% of the total electorate who decided to leave, and the whole of Scotland voted to remain heavily - does this legitimise the government to carry out Brexit? Thus, when we look at the matter from a different perspective, should such an important constitutional matter be given to the (sometimes) uninformed public, or should it require a 2/3 representative majority approval as it is in America?