No, but they share logic as the foundation. A SAT solver merely solves a series of Boolean equations, typically in conjunctive normal form. Prolog has deduction capabilities that go far beyond that, where you can reason over a tree data structure, computing various parts of it according to a set of constraints. A SAT solver is not Turing complete. Prolog is. You could use Prolog to write a SAT solver (though it wouldn’t be very competitive with solvers written in C or other languages).