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 $i\varepsilon$-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.