We construct the causal fermion system for globally hyperbolic spacetimes starting in the framework of algebraic quantum field theory. The fermionic projector is identified with the one-particle density operator of a quasi-free Hadamard state. The ultraviolet regularization is built into the fermionic projector via a chart-independent iε regularization scheme. The continuum limit analysis is developed in globally hyperbolic spacetimes. It is shown that the Euler–Lagrange equations of the causal action principle are satisfied in this setup if and only if the coupled Einstein–Dirac equations hold.