As a promising new technology for green communication, backscatter communication has attracted wide attention in academics and industry. This paper studies the resource allocation problem for an unmanned aerial vehicle (UAV)-assisted backscatter communication network. The UAV acts as an airborne mobile base station and broadcasts signals to the ground backscatter devices (BDs), which transmit the data signals to the backscatter receiver (BR) in a backscattered manner. Specifically, the ground-based BDs communicate with the BR using a dynamic protocol based on time division multiple access. Considering the fairness among BDs, we aim to investigate maximizing the minimum (max–min) rate of the proposed network by jointly optimizing backscatter device scheduling, reflection coefficient, UAV’s power control, and UAV’s trajectory. The optimization problem is a non-convex problem, which is challenging to obtain the optimal solution. Therefore, we propose an efficient iterative algorithm to decompose the optimization problem into four subproblems by the block coordinate descent method. The variables are alternatively optimized by the interior point method and successive convex approximation techniques in each iteration. Finally, simulation results show that the max–min rate of the system obtained by the proposed scheme outperforms other benchmark schemes.