The usual suggestion is not to ignore the vote, but to put it back up for a vote. Ask people if, now that they've seen the difficulty of Brexit, they still want to do this. If the people actually want the Brexit that exists in real life, they'll reaffirm. It's still democratic by any reasonable definition of the word.
First of all, two votes is not "again and again" in any sense of the phrase. I take your point, but nobody is suggesting referendums without end.
Second of all, bringing something up for a vote more that once is not like ignoring the vote. Rejected proposals are very often brought back up, maybe with some retooling, maybe just in hopes that the passage of time has made them more palatable. For example, Quebec voted on its equivalent of Brexit twice, and though it was a bit of a joke, it wasn't generally viewed as shady.
It is possible to use additional referendums as a stalling tactic, and if you kept doing it forever, you would effectively be ignoring the vote. But again, nobody is suggesting that. Putting it up for another vote now that the public has more information would literally just be democracy.
