But ...
There has already been a second referendum!
When we voted in the first referendum (in 1975, as I recall), we voted to remain -- by a margin of more than two-to-one. (I voted then, as did many of my friends and family.)
It would certainly be unreasonable, not to say blatantly contradictory, for 'leavers' and their apologists now to claim another referendum would be 'undemocratic', or as OP suggests, 'unjustified'.
If, as OP and others try to claim, a second referendum should be considered undemocratic, the referendum in 2016 was undemocratic.
To even approach being reasonable, we need either to have another referendum ... or simply to annul Article 50 on the grounds that the 2016 vote was undemocratic (and anyway in the end did not contradict the first referendum either on number of votes or proportion of votes cast).
Overall, in the two referenda we have already had on this issue, many more votes were cast in favour of remain than the contrary.
We would be wholly justified, democratically and reasonably, in annulling Article 50 based just on this fact alone. Unfortunately, Cameron, Mogg, Johnson and other Etonians and their associates ... oh, and of course May ... ...
Practically speaking, yes we need another referendum, confirmatory vote, call it what you will.
It's one-all, going into extra time! Vote Remain! We love EU!