Despite extensive research on how patient engagement behaviors (PEBs) are facilitated though explicit technical interventions in medical services, research on the encouragement of PEBs from the perspective of the service process is lacking. This study explores how functional quality dimensions (responsiveness, empathy, surroundings, and access) affect PEBs (compliance and loyalty) through a two-channel psychological mechanism (trust and satisfaction). This study tests the proposed model using survey data from two public hospitals in southeastern China and employs the partial least square (PLS) technique of structural equation modeling (SEM) to analyze the data. The results show that service providers’ responsiveness, empathy, and access affect patient compliance and loyalty through patient satisfaction; however, the effect of surroundings is not significant. The responsiveness and empathy of service staff affect PEBs through patient trust. Considering the high-contact professional nature of medical services, we call for more efforts toward improving service processes rather than simply relying on technical interventions. Specifically, hospitals and contact employees should devote time and effort to functional quality management in three dimensions, namely responsiveness, empathy, and access, to secure patient trust and satisfaction.