Urban Air Mobility (UAM) offers a solution to current traffic congestion by providing on-demand air mobility in urban areas. Effective traffic management is crucial for efficient operation of UAM systems, especially for high-demand scenarios. In this paper, we present a centralized traffic management framework for on-demand UAM systems. Specifically, we provide a scheduling policy, called VertiSync, which schedules the aircraft for either servicing trip requests or rebalancing in the system subject to aircraft safety margins and energy requirements. We characterize the system-level throughput of VertiSync, which determines the demand threshold at which passenger waiting times transition from being stabilized to being increasing over time. We show that the proposed policy is able to maximize throughput for sufficiently large fleet sizes. We demonstrate the performance of VertiSync through a case study for the city of Los Angeles, and show that it significantly reduces passenger waiting times compared to a first-come first-serve scheduling policy.