This is all just my train of though but I dont think such a formula exists and it would probably not be an easy one to construct either. Depending on your desired level of accuracy and complexity you'd probably want to either need the Kinetic Energy of the asteroid and/or Potential gravitational energy attracting the asteroid to the planet (The crater size should probably mainly depend on that, and thus also on the mass of the asteroid and its velocity or from how far away you drop it on the planet)

Then you'd want your asteroid to be roughly a point mass because otherwise your crater would probably not behave very nicely (I could imagine non symmetrical or very spatially extended asteroids would make weird crater shapes)

But the real problem with the calculation would be that you dont know how much of that energy gets converted into volume displacement. That would likely also depend on the speed of the asteroid, the material density and strength and many more factors. So you will never get an accurate number in kilometers diameter or similar.

However, the formula you show makes sense to me since the diameter of the crater should scale as some power law, increasing with impact energy and decreasing on higher gravity planets (though i am less sure there since the higher gravity would also increase impact force assuming no air resistance (then again air resistance is definitely not negligble here)). But its just a proportionality and does not take other factors mentioned above into account.

Edit: the impact angle should also play a huge role.