Can the gravitational constant $G$ be calculated theoretically?

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).

No, as commented on above.

Worse, we don't know its value very well.

There are efforts underway to measure $G$ more accurately, as reported in Nature earlier this month:

It is one of nature’s most fundamental numbers, but humanity still doesn’t have an accurate value for the gravitational constant. And, bafflingly, scientists’ ability to pinpoint G seems to be getting worse. This week, the world’s leading gravity metrologists are meeting to devise a set of experiments that will try to set the record straight. This will call for precision measurements that are notoriously difficult to make — but it will also require former rivals to work together...

The gravitational constant can only be determined experimentally. It's empirically derived. There is no physical law we know of that dictates the strength of $G$. Rather, we have the observation that gravitational force is:

$F_g = G \frac{m_1 m_2}{r^2}$

Since we can know the other four numbers, we must solve for $G$ experimentally. For a brief background: http://en.wikipedia.org/wiki/Gravitational_constant#History_of_measurement