$G$ is just a constant of proportionality to get the units right (so that when $m_1$ and $m_2$ are in kilograms and $r$ is in meters you get a force in Newtons rather than wingdingalings or something really weird). Indeed cosmologists like to work in a system of units where $G = c = 1 \text{ (dimensionless)}$, and particle physicists like to work in units where $c = \hbar = 1 \text{ (dimensionless)}$, you can even set all three of these numbers to 1 if you like.
In your game implementation you have broad freedom to choose the scale of the masses and the time intervals over which to evaluate stuff, so obviously you can ditch the constant if you want to. Now, if you want to model the solar system and to use the masses you find in a table for the planets and the sun, you'll have to include the constant.
This has nothing to do with the size of the constant however, because it is a multiplicative coefficient.