In this paper we present an automatic and language-independent program verification approach based on symbolic execution. The specification formalism we consider is Reachability Logic, a language-independent logic that constitutes an alternative to Hoare logics. Reachability Logic has a sound and relatively complete deduction system, which offers a lot of freedom (but no guidelines) for constructing proofs. Hence, we propose symbolic execution as a strategy for proof construction. We show that, under reasonable conditions on the semantics of programming languages, our symbolic-execution based Reachability-Logic formula verification is sound. We present a prototype implementation of the resulting language-independent verifier as an extension of a generic symbolic execution engine that we are developing in the K framework. The verifier is illustrated on programs written in languages also formally defined in K.