In this paper, the problem of resource management is studied for a network of wireless virtual reality (VR) users communicating using an unmanned aerial vehicle (UAV)-enabled LTE-U network. In the studied model, the UAVs act as VR control centers that collect tracking information from the VR users over the wireless uplink and, then, send the constructed VR images to the VR users over an LTE-U downlink. Therefore, resource allocation in such a UAV-enabled LTE-U network must jointly consider the uplink and downlink links over both licensed and unlicensed bands. In such a VR setting, the UAVs can dynamically adjust the image quality and format of each VR image to change the data size of each VR image, then meet the delay requirement. Therefore, resource allocation must also take into account the image quality and format. This VR-centric resource allocation problem is formulated as a noncooperative game that enables a joint allocation of licensed and unlicensed spectrum bands, as well as a dynamic adaptation of VR image quality and format. To solve this game, a learning algorithm based on the machine learning tools of echo state networks (ESNs) with leaky integrator neurons is proposed. Unlike conventional ESN based learning algorithms that are suitable for discrete-time systems, the proposed algorithm can dynamically adjust the update speed of the ESN's state and, hence, it can enable the UAVs to learn the continuous dynamics of their associated VR users. Simulation results show that the proposed algorithm achieves up to 14% and 27.1% gains in terms of total VR QoE for all users compared to Q-learning using LTE-U and Q-learning using LTE.