There's no international law that says a country has to let a part of it secede.
I'm not sure about this, and I doubt you are either. International law is complicated and routinely ignored.

I'll think you'll find if the UK doesn't at least start making noises towards rejoining the EU in the next five years, there will be another Scottish referendum with a different outcome, and possible a N. Irish one too. Then we'll see how the UK responds. Most countries are not keen to see parts of themselves break away.
Are you forgetting we already allowed Scotland to vote on this matter? The UK doesn't want it, but the UK might not be able to stop it.