Would it be true that if you double the distance of an astronaut from a planet, the gravitational pull between them would be half as strong?

Yes

Explanation:

Newton's law of universal gravitation is usually stated that every particle attracts every other particle in the universe with a force which is directly proportional to the product of their masses (m1 and m2) and inversely proportional to the square of the distance between their centers (r).

F = Gm1m2/r²

This is a general physical law derived from

empirical observations by what Isaac Newton called inductive reasoning.

when distance is doubled the gravitational force will be reduced by quarter not half.

Yes, that would be the case, because the gravity is pulling you towards the planet, so doubling the the distance cuts the gravitational pull between them in half.