Matt, you're not correct there....EU directive requires UK legislation through an act of parliament to implement them.
The EU referendum wording was "Should the UK remain a member of the European Union?" iirc. The outcome of the referendum was that no, the UK should not.
That's fine. But that is all that the referendum asked.
What would not be fine is for May, together with Fox, Davis and Boris to decide (dictate) what path is taken in related areas such as trade - e.g. going for an EEA option or WTO rules option - without proper recourse to the democratic institution of the UK i.e. parliament. Quite how anyone (in the press, on this forum, wherever) can claim the judgement regarding A50 is anti-democratic and at the same time claim the outcome of the referendum was about sovereignty and democracy is baffling to say the least.