Voltage and frequency control in an islanded ac microgrid (ImG) amounts to stabilizing an a priori unknown ImG equilibrium induced by loads and changes in topology. This article puts forth a unified control framework, which, while guaranteeing such stability, allows for modular ImGs interconnecting multiple subsystems, that is, dynamic RLC lines, nonlinear constant impedance, current, power (ZIP) and exponential (EXP) loads, and inverter-based distributed generation units (DGUs) controlled with different types of primary controllers. The underlying idea of the framework is based on the equilibrium-independent passivity (EIP) of the ImC subsystems, which enables stability certificates of ImG equilibria without explicit knowledge of these equilibria. In order to render DGUs EIP, we propose a decentralized controller synthesis algorithm based on port-Hamiltonian systems (PHSs). We also show that EIP, being the key to stability, provides a general framework, which can embrace other solutions available in the literature. Furthermore, we provide a novel argument based on LaSalle's theorem for proving asymptotic voltage and frequency stability. Finally. we analyze the impact of actuator saturation on the stability results by exploiting the inherent EIP properties of the PHS DGU model. Theoretical findings are backed up by realistic simulations based on the medium-voltage CIGRE benchmark network.