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A simple renormalization theory of plasma particle interactions is proposed. It primarily stems from generic properties of equilibrium distribution functions and allows one to obtain the so-called generalized Poisson-Boltzmann equation for an effective interaction potential of two chosen particles in the presence of a third one. The same equation is then strictly derived from the Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy for equilibrium distribution functions in the pair correlation approximation. This enables one to construct a self-consistent chemical model of partially ionized plasmas, correctly accounting for the close interrelation of charged and neutral components thereof. Minimization of the system free energy provides ionization equilibrium and, thus, permits one to study the plasma composition in a wide range of its parameters. Unlike standard chemical models, the proposed one allows one to study the system correlation functions and thereby to obtain an equation of state which agrees well with exact results of quantum-mechanical activity expansions. It is shown that the plasma and neutral components are strongly interrelated, which results in the short-range order formation in the corresponding subsystem. The mathematical form of the results obtained enables one to both firmly establish this fact and to determine a characteristic length of the structure formation. Since the cornerstone of the proposed self-consistent chemical model of partially ionized plasmas is an effective pairwise interaction potential, it immediately provides quite an efficient calculation scheme not only for thermodynamical functions but for transport coefficients as well.