This paper introduces GCNSplit, a streaming graph partitioning framework capable of handling unbounded streams with bounded state requirements. We frame partitioning as a classification problem and we employ an unsupervised model whose loss function minimizes edge-cuts. GCNSplit leverages an inductive graph convolutional network (GCN) to embed graph characteristics into a low-dimensional space and assign edges to partitions in an online manner. We evaluate GCNSplit with real-world graph datasets of various sizes and domains. Our results demonstrate that GCNSplit provides high-throughput, top-quality partitioning, and successfully leverages data parallelism. It achieves a throughput of 430K edges/s on a real-world graph of 1.6B edges using a bounded 147KB-sized model, contrary to the state-of-the-art HDRF algorithm that requires > 116GB in-memory state. With a well-balanced normalized load of 1.01, GCNSplit achieves a replication factor on par with HDRF, showcasing high partitioning quality while storing three orders of magnitude smaller partitioning state. Owing to the power of GCNs, we show that GCNSplit can generalize to entirely unseen graphs while outperforming the state-of-the-art stream partitioners in some cases.