An exact treatment of the van der Waals interaction between two spherical dielectric bodies possessing purely classical degrees of freedom is presented. The spheres are described by multipole expansions of their fluctuating charge distributions, and the correlation between the fluctuations are taken into account using classical electrostatics and statistical mechanics. The presented approach avoids both the assumption of pairwise additivity of Hamaker theory and the implicit linear response assumption of Lifshitz theory. The resulting equations are solved numerically for D/a >= 0.01, where a is the radius of the spheres and D is their minimum separation, for a system with epsilon = 80, and the results are compared to the analytical Hamaker formula with a Hamaker constant calculated from Lifshitz theory.