While the constant itself is not dependent on anything else, its effect on the universe is dependent. For example, $G$ is a sort of ratio that relates the force between two objects at a given distance and of two certain masses. $G$ does not affect the electrical force between two charged particles - that's Coulomb's constant; it does not affect the time it takes a massless particle to travel a certain distance - that's the "speed of light," $c$. So really, in a way, it's not quite independent of everything, because in order to have $G$ you have to have masses, distances and forces; otherwise the number itself is useless.
If the gravitational constant were dimensionless, in fact, it would also be meaningless because there would be no way to keep it actually constant. Since it is a ratio, you have to determine what things it is a ratio between; in this case, again, those values are mass, distance and gravitational "force." If there were no units, you couldn't usefully calculate how gravity affects objects; you'd have a number, but no way to use it - really, no way even to calculate the constant to begin with, now that I think about it. While there are meaningful dimensionless quantities, famously the fine structure constant and (more mundanely) percentages for example, they can only arise naturally in cases where the things you are comparing are of the same nature. For example, "how many meters are in a light-year" is a dimensionless quantity, because it is a ratio of two units of distance. Comparing this to $G$ it's easy to see why the constant is not dimensionless; it compares units of different natures, specifically force, distance and mass, none of which can be converted into any of the others unless by a constant such as $G$ which, well, that's pretty much the whole reason $G$ exists at all.