It's not anti-democratic to protest, or to march, or to complain about a decision you think is wrong. Nor is there much point in asking the question about what would happen if the positions were reversed. I'll tell you what would happen: the narrowly defeated Brexit supporters would certainly not think "Oh, I guess staying in the EU was the right thing to do after all, mine eyes have been opened and I see the light now that the EU is the beacon of freedom and prosperity rather than, as I argued passionately yesterday, the shackle on our access to the world and the noose around our necks."
And why not? Brexit supporters voted for a cause they believed in, and in the event of a narrow victory in Remain's favour would it not provide encouragement that the argument could be won in the near future, or under slightly different circumstances? And here's the thing, too, why shouldn't you keep pushing for what you believe in? You might accept the legitimacy of the result, but you would certainly not accept the correctness of it.
The same is true for Remainers. I can't speak for all who voted to leave the EU but I certainly still believe I made the right decision, even if the country as a whole disagreed with me.
None of this should change the general direction the country is set in now, which is Brexit, but I'm getting tired of seeing Brexit supporters conflating protest at the correctness of a decision with protest at the legitimacy of it.