Jay, maybe you also missed the fact that it was a Leaver who started the petition asking for a second referendum,
which has now been signed by four million people and is due to be debated on 5 September.
There is no symmetry between whether there was a Leave or Remain win. There should always be a presumption in favour of the status quo and an obstacle to change. This has already been observed here by comparison to the US Constitution, for example, but since this is a tech site let me put it in engineering terms.
The thermostat on your central heating doesn't turn it on as soon as the room goes 0.1 degree below the set level, because to do so would damage the system. Anyone who argues that one vote over 50% should be enough to trigger drastic change must equally accept that if two people change their mind then there must immediately be a drastic reversal. Clearly that would be stupid. Ergo, making the drastic change in the first place on such a flimsy margin is stupid.
PS don't forget the March for Europe on 3 September.