This paper considers a closed queueing network model of ridesharing systems such as Didi Chuxing, Lyft, and Uber. We focus on empty-car routing, a mechanism by which we control car flow in the network to optimize system-wide utility functions, e.g. the availability of empty cars when a passenger arrives. We establish both process-level and steady-state convergence of the queueing network to a fluid limit in a large market regime where demand for rides and supply of cars tend to infinity, and use this limit to study a fluid-based optimization problem. We prove that the optimal network utility obtained from the fluid-based optimization is an upper bound on the utility in the finite car system for any routing policy, both static and dynamic, under which the closed queueing network has a stationary distribution. This upper bound is achieved asymptotically under the fluid-based optimal routing policy. Simulation results with real-world data released by Didi Chuxing demonstrate the benefit of using the fluid-based optimal routing policy compared to various other policies.