This paper presents a quasi-decentralized networked control methodology for the robust stabilization of process networks whose constituent subsystems communicate over a shared resource-constrained communication medium and are subject to time-varying external disturbances. The objective is to stabilize the plant and attenuate the influence of the disturbances while keeping the communication to a minimum in order to reduce the unnecessary utilization of network resources. Both state- and output-feedback control problems are considered. Initially, a local robust feedback controller is synthesized for each unit to account for the effect of the disturbances. Then, the exchange of information between the distributed control systems is reduced by embedding, within each control system, a set of dynamic models that provide forecasts of the evolution of the plant states when measurements are not transmitted through the shared network, and updating the state of each model when communication is re-established at discrete time instances. By analyzing the resulting combined discrete-continuous closed-loop system, a necessary and sufficient condition for robust stability of the networked closed-loop plant is obtained. The stability condition can be used to explicitly characterize the interplay between the update period, the degree of plant model mismatch, the selection of the controller design parameters, the size of the disturbances, and the size of the achievable degree of disturbance attenuation. Finally, the implementation of the robust networked control structure is demonstrated through an application to a chemical plant example.