The coupling-parameter equations for spin-pair correlation functions are examined for field-free Ising models with coordination number four (including both two- and three-dimensional cases). By means of sets of identities discovered by Fisher, it proves possible to eliminate exactly all higher order correlation functions from the equations, in the event that only nearest-neighbor sites interact. As a result, one can rigorously show for this class of Ising models that each spin-pair correlation function (one spin partially coupled) is a linear combination of two independent functions of the coupling parameter, and that spatial dependence occurs only through their multiplicative factors. Only one relation between these spatial factors is available for each site-pair separation distance, so that the Ising problem in its usual interpretation (coupling-parameter unity) is not yet rigorously soluble by this approach. However, by using the correlation function for the fully coupled case itself as a second set of constraints, exact results can be obtained explicitly for the pair correlation function, and hence the solution thermodynamics, for dilute impurities coupled with arbitrary strength to their nearest neighbors.