We design, develop, and evaluate an atomic- and lock-free GPU implementation of the push-relabel algorithm in the context of finding maximum cardinality matchings in bipartite graphs. The problem has applications on computer science, scientific computing, bioinformatics, and other areas. Although the GPU parallelization of the push-relabel technique has been investigated in the context of flow algorithms, to the best of our knowledge, ours is the first study which focuses on the maximum cardinality matching. We compare the proposed algorithms with serial, multicore, and manycore bipartite graph matching implementations from the literature on a large set of real-life problems. Our experiments show that the proposed push- relabel-based GPU algorithm is faster than the existing parallel and sequential implementations.