How to calculate bond angle of simple molecules quantitatively?
This has a simple reason: To the best of my knowledge there is no quantitative way to predict bond angles without resorting to computational chemistry.

This is because the entire subject of predicting bond angles is a lot more tricky than it was displayed to you. Take phosphane and hydrogen sulphide, the heavy analogues to ammonia and water. Their bond angles are nowhere near $109^\circ$ but much closer to $90^\circ$.

If you dig deeper into the matter, you find that ammonia and water are more the exception than the rule, that the VSEPR model fails spectacularly on the long run, and that their geometries are actually finely balanced between an ideal $90^\circ$ angle and the hydrogen-hydrogen repulsion.

In fact, not even with computers can you calculate the bond angle directly with a certain set of axiomic assumptions. Rather, computational chemistry would empirically calculate energies for different angles and deem the angle that has the lowest energy to be ‘correct’ and ‘calculated’.

So no simple quantitative rule exists nor will it exist.



Note: There are, of course, semi-quantitative rules like the one mentioned by Ringo in a comment. But I don’t consider them quantitative, because they still contain approximations.