People seem confused over the word "democracy".
We live in a democracy, so everything before the referendum was "democratic" and it would have also been "democratic" to not have a referendum. There's nothing to say that a referendum is more democratic than the way we normally arrive at decisions. You might ask yourself, in a future where voting was a lot more efficient, whether you'd be happier if every single issue was electronically decided by the people rather than your elected representatives. Sounds like a recipe for chaos to me ...
Reaching decisions by referendum is no more or less democratic than reaching decisions through a representative democracy - especially when lies were told throughout the referendum campaign (justified by "They were on both sides" - as if it cancels out!), electoral spending limits were breached it appears, and the percentage of the electorate that actually voted Leave was 37%.
This is not democracy. It's people whose ideology was troubled by the EU, throwing their toys out of the pram, kicking up a fuss, getting their referendum through a weak and feeble PM, lying and cheating their way to a "victory" that wouldn't be sufficient to change the rules at your average golf club, then insisting that victory meant one thing and only one thing as if the closeness of it and the 37% actual Leave vote meant nothing at all; as if democracy was not "the will of the people" as a whole but "the will of the 37%". That's closer to ochlocracy than democracy. Anyway, onwards into our golden future ...