We find the emergence of strong correlations and universality on the approach to the quantum critical points of a two-impurity Anderson model. The two impurities are coupled by an interimpurity exchange interaction J and direct interaction U-12 and are hybridized with separate conduction channels. The low energy behavior is described in terms of renormalized parameters. We show that on the approach to the transitions to a local singlet and a local charged ordered state, the quasiparticle weight factor z -> 0, and the renormalized parameters can be expressed in terms of a single energy scale T*. The values of the renormalized interaction parameters in terms of T* can be predicted from the condition of continuity of the spin and charge susceptibilities, and correspond to strong correlation. These predictions are confirmed by the numerical renormalization group calculations, including the case when the on site interaction U = 0.