Boltzmann's constant is not "fundamental" in the same sense as $c$ or $G$. Rather, it is an artifact of measuring temperature in units of kelvins rather than joules. Other non-fundamental constants can often be expressed compactly in terms of other more fundamental constants. (For example, the Rydberg constant can be expressed as $R_\infty=\frac{m_ee^4}{8\epsilon_o^2h^3c}$.)
So it would stand to reason that $k_B$ can be calculated from other more fundamental constants as well (even if not from a nice compact equation like the one given for $R_\infty$ above). In particular, $k_B$ essentially converts between units of joules and units of kelvins. The kelvin is defined as 1/273.16 the temperature difference between absolute zero and the temperature of the triple point of water. Water is made of H2O molecules, which obey the laws of quantum mechanics. So I would expect there to be a (perhaps ugly) expression for $k_B$ in terms of statistical properties of water, which in turn should reduce to fundamental constants like $h, c, m_e,$ etc.
1. Does such an expression for $k_B$ exist (at least in principle)?
2. Could $k_B$ be calculated theoretically from computer simulations of water, and if so, has anyone done it?