Well, natural deduction is one specific form of proof theory (it's a kind of proof system), but there are others. Moreover, proof theory and model theory are independent (neither relies on the other) and various logicians would say that we don't really need one of them, though they would disagree on which. Proof theorists tend to be skeptical of the necessity of model theory, model theorists tend to be skeptical of the necessity of proof theory. Mathematicians tend to be skeptical of both.

But yes, to the degree that any such theoretical parts are part of predicate logic, they are already part of propositional logic. The former is an extension of the latter. Perhaps similar to how higher-order logic (simple type theory) extends or generalizes both propositional and first-order predicate logic.