= Solution
Under uniform geometric scaling, write $\mathbf r'_i=\lambda\mathbf r_i$. Each volume element contributes a factor $\lambda^3$, while the inverse-sixth-power separation contributes $\lambda^{-6}$. The double integral therefore scales as $\lambda^3\lambda^3\lambda^{-6}=1$, and
$$
\boxed{V'_{\rm vdW}=V_{\rm vdW}.}
$$
This assumes unchanged material <Hamaker constant>, pairwise-additive nonretarded interactions and the same intervening medium. It is a simultaneous scaling of both body sizes and their separation, not merely moving fixed bodies apart.
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