This paper develops a fast mixed-integer convex programming (MICP) framework for multi-robot navigation by combining graph attention networks and distributed optimization. We formulate a mixed-integer optimization problem for receding horizon motion planning of a multi-robot system given surrounding obstacles. To solve the resulting multi-agent MICP problem in real-time, we propose a framework that utilizes heterogeneous graph attention networks to learn the latent mapping from problem parameters to optimal binary solutions and a distributed proximal alternating direction method of multipliers (ADMM) algorithm for solving the convex continuous optimization problem. We demonstrate the effectiveness of the proposed framework through experiments on a robotic testbed.