I know that sounds a bit hyperbolic but really, if you are not happy to accept the results of our democratic process, you need to say what you are going to replace our democracy with when it has been ripped to shreds.
If you try to blend direct democracy with representative democracy, things are going to get messy. By introducing this by-popular-request, one-night-only, ain't-we-good-to-you election-bait of a nationwide referendum on a contentious subject, Cameron instantly undermined elected representatives, parliamentary sovereignty and the judiciary ('the people have spoken!' = if you act contrary to the 52%, you can be defined as 'enemies of the people').
Now direct democracy's done its bit, we're tasking these elected representatives with enacting the people's will - except the practical application of the decision, which was offered as a simple yes/no, has a thousand different permutations. FoM, passporting rights, the single market, subsidies, rights of EU residents of the UK... I could fill an entire page with these issues, and - judging from everything I've heard and read from Brexit voters - there would be a huge diversity of opinion. How are they supposed to divine the people's will?
Ok - fine. Not a disaster. It's a one-off referendum, the people get to make the big decision, and then we'll return all authority to our elected representatives to hammer out the details. That should be an excellent way of making sure a wide variety of concerns are heard, including those of the 48%, who are still, obviously, people, and constituents of the elected representatives. If the will of the 52% counts for so much, then the 48% must be acknowledged.
Except our own politicians then descend into an undignified, self-interested power-struggle. The opposition are in an absolute shambles, and the right-wing end up with a leader who is set on moving forward with a hard Brexit, with as little debate or discussion as possible. She has to be prodded, at every stage, to allow fundamental parts of the parliamentary process to occur.
This is democracy at its most ineffectual and dangerous; when it is not particularly respecting our own democratic system of checks and balances, but any dissent - lawful dissent, such as a peaceful protest march or a fecking court case - can be shouted down as 'an assault to democracy' and 'disrespecting the people's will'. As is typified by your post. People are 'ripping democracy to shreds' by engaging in their democractic right to legal civic protest.
I'm glad you have such a full, unconditional and absolute respect for democracy. How would you stand on a 52/48 referendum to ban abortion? 52/48 we'll bring in US-style capital punishment? To basically any issue which runs contrary to your own personal conscience?