Assuming Wikipedia isn't far from the truth in the very first paragraph on the topic:
National referendums can be permitted by an Act of Parliament and regulated through the Political Parties, Elections and Referendums Act 2000, but they are by tradition extremely rare due to the principle of parliamentary sovereignty meaning that they cannot be constitutionally binding on either the Government or Parliament, although they usually have a persuasive political effect.
So no referendums can be truly legally binding in the UK (at national level)... unless the UK's party unwritten constitutional arrangements get a substantially different interpretation.
Or at least that's the received wisdom. As far as I can tell from reading the Act authorizing the AV referendum, that one was binding basically... and had simple majority specified as well, but Parliament could have still changed its mind later and could have (in theory) abrogated that law (although it didn't).
The Minister must make an order bringing into force section 9, Schedule 10 and Part 1 of Schedule 12 (“the alternative vote provisions”) if—
(a) more votes are cast in the referendum in favour of the answer “Yes” than in favour of the answer “No”, and
(b) [impenetrable legalese about some Order]
(2) If more votes are not cast in the referendum in favour of the answer “Yes” than in favour of the answer “No”, the Minister must make an order repealing the alternative vote provisions.
FullFact comments on that one
A UK referendum will only have the force of law if the Act setting it up says so. In practical terms this would mean someone would be able to go to court to make the government implement the result. The Alternative Vote referendum in 2011, for example, was legally binding in this way.
So, no there's no general law like you ask, but there is one example of a previous UK referendums that was "more binding" than the Brexit one; it also had the majority rule specified in the Act authorizing it (unlike the Brexit one).
As for why the UK isn't doing something else entirely... well that would involve a lot of speculation. Perhaps the "first past the post" political culture (see rejection of AV) extends to referendums.