What is the connection between Planck's constant and gravity? Why is the Planck scale the natural scale for quantum gravity? I would have though the scale would be related to G, not h.
There is a difference between Planck's constant $h$ and the "Planck scale" as it is most often used colloquially, the latter referring to the Planck length, Planck time, and Planck mass. Planck's constant, as you note, has no direct connection to gravity and can be derived experimentally by studying black body spectra, as one example. The "Planck scale" involves gravity by design: it refers to the physical scales at which both general relativity and quanutum mechanics are relevant, and so involves both the gravitational constant $G$ and the Planck constant $h$.