I doubt very much if a referendum could be run on legal grounds. Even the petition would require retrospective legislation, although even that hasn't been beyond the government .
However, the petition has just collected 2 million signatures. Is any government really going to be happy knowing that there are 2 million seriously pi$$ed off voters in the country? This referendum was held partly because UKIP polled over 4 million, so it's serious number of grumpy voters.
If I were Dave, I'd hold a new referendum. I'd introduce a form of list system for elections, which would inevitably mean some UKIP MPs, but that's fine. Then I'd hold a new election, which the Conservatives would probably win. I'd appoint Boris, IDS, Gove, etc as governors of some far-flung islands and get on with the job of listening to people and getting this country back on track.