gravity - Can the gravitational constant $G$ be calculated theoretically?

We know all that the gravitational constant is $$G=6.67428±0.00067\times 10^{-11}\mathrm{m^{3} \:kg^{-1} s^{-2}}.$$ But can we calculate it theoretically?

Answer

You can't calculate the numerical value of Newton's constant from the first principle because it is a dimensionful constant – it has units – so the numerical value depends on the magnitude of the units. And because e.g. the kilogram is defined as the mass of a platinum prototype hosted by a French chateau (the kilogram has the "least objective" definition so far), it's clear that a "pure calculation" can't know how large the kilogram is, which also means that it can't determine the numerical value of Newton's constant which depends on the definition of a kilogram.

In other units, e.g. Planck units, people often set $G=1$ or $G=1/8\pi$. In that case, the constant may be calculated – I just did it. If one uses such units, there are other – dimensionless, and therefore potentially calculable from the first principles – constants of Nature such as the electron mass (in the unit of the Planck mass). String theory is the only framework in physics that allows one to calculate all these continuous dimensionless universal constants of physics. One may prove that for a given (stabilized) compactification of string theory, all of these constants are fully determined. In practice, physicists can't do that yet because they don't know how to choose the right compactification (which is just a discrete amount of currently uncertain information that must be inserted to the calculation).