Foundational verification considers the functional correctness of real programming languages with formalized semantics and uses proof assistants (e.g., Coq, Isabelle) to certify proofs. The need for verifying real languages compels it to involve expressive Separation Logics (SLs) that exceed the scopes of well-studied automated proof theories, e.g., symbolic heap. Consequently, automation of SL in foundational verification relies heavily on ad-hoc heuristics that lack a systematic meta-theory and face scalability issues. To mitigate the gap, we propose a theory to specify SL predicates using abstract algebras including functors, homomorphisms, and modules over rings. Based on this theory, we develop a generic SL automation algorithm to reason about any data structures that can be characterized by these algebras. In addition, we also present algorithms for automatically instantiating the algebraic models to real data structures. The instantiation reuses the algebraic models of component structures and preserves their data abstractions. Case studies on formalized C semantics show our algorithm can instantiate the algebraic models automatically for a variety of complex data structures. Our algorithm is able to automate the verification of complex programs in Isabelle/HOL with minimal manual effort. This indicates the automatically instantiated reasoners from our generic theory show similar results to the state-of-the-art systems made of specifically crafted reasoning rules. The presented theories, proofs and the verification framework are formalized in Isabelle/HOL.