A rigorous approach is proposed to calculate the electrostatic forces among an arbitrary number of solvated molecules in ionic solution determined by the linearized Poisson-Boltzmann equation. The variational principle is used and implemented in the frame of a boundary element method (BEM). This approach does not require the calculation of the Maxwell stress tensor on the molecular surface, therefore it totally avoids the hypersingularity problem in the direct BEM whenever one needs to calculate the gradient of the surface potential or the stress tensor. This method provides an accurate and efficient way to calculate the full intermolecular electrostatic interaction energy and force, which could potentially be used in Brownian dynamics simulation of biomolecular association. The method has been tested on some simple cases to demonstrate its reliability and efficiency, and parts of the results are compared with analytical results and with those obtained by some known methods such as adaptive Poisson-Boltzmann solver.