We present the design, development, and experimental validation of BlueME, a compact magnetoelectric (ME) antenna array system for underwater robot-to-robot communication. BlueME employs ME antennas operating at their natural mechanical resonance frequency to efficiently transmit and receive very-low-frequency (VLF) electromagnetic signals underwater. We outline the design, simulation, fabrication, and integration of the proposed system on low-power embedded platforms, focusing on portable and scalable applications. For performance evaluation, we deployed BlueME on an autonomous surface vehicle (ASV) and a remotely operated vehicle (ROV) in open-water field trials. Ocean trials demonstrate that BlueME maintains reliable signal transmission at distances beyond 700 meters while consuming only 10 watts of power. Field trials show that the system operates effectively in challenging underwater conditions such as turbidity, obstacles, and multipath interference -- conditions that generally affect acoustics and optics. Our analysis also examines the impact of complete submersion on system performance and identifies key deployment considerations. This work represents the first practical underwater deployment of ME antennas outside the laboratory and implements the largest VLF ME array system to date. BlueME demonstrates significant potential for marine robotics and automation in multi-robot cooperative systems and remote sensor networks.