The question is the title. But I'm quite doubtful if this question is meaningful or not. Since this constant is obtained by experiment, we can never know its exact value, unlike $π$ or $e$. Is it correct?
It depends on your system of units. In SI units G is an experimentally determined number known only to a certain precision. In Planck units it is 1, which is clearly rational. Other systems of units will vary.
This is generally true of most universal dimensionful constants.
According to our current theoretical paradigms (general relativity, quantum field theory), it's an arbitrary real number.
In string theory, in each model that resembles the real world, it should have a uniquely determined value. I don't know what kind of numbers those values are, but they are likely to contain contributions from square roots and transcendental functions, and not just ratios of integers.