Looks like a homework problem.
Just knowing the value of G and the mass of the Earth is not enough. You need to know at least another value, such as the orbital period of the Moon. For example, if you know that the Moon takes 28 days to complete its orbit around the Earth, you can calculate the centripetal acceleration as a function of r. The acceleration due to gravity equals the centripetal acceleration. Take your equation for r, substitute the centripetal acceleration (function of r) for g, and solve for r. This is only very approximate, because in fact the Earth and moon orbit around their common center of mass, so the radius of the moon's orbit is smaller than the distance from Earth to Moon. If you know the mass of the Moon you can calculate whee the common center of mass is, and make the appropriate correction.