Current maritime communications mainly rely on satellites having meager transmission resources, hence suffering from poorer performance than modern terrestrial wireless networks. With the growth of transcontinental air traffic, the promising concept of aeronautical ad hoc networking relying on commercial passenger airplanes is potentially capable of enhancing satellite-based maritime communications via air-to-ground and multi-hop air-to-air links. In this article, we conceive space-air-ground integrated networks (SAGINs) for supporting ubiquitous maritime communications, where the low-earth-orbit satellite constellations, passenger airplanes, terrestrial base stations, ships, respectively, serve as the space-, air-, ground- and sea-layer. To meet heterogeneous service requirements, and accommodate the time-varying and self-organizing nature of SAGINs, we propose a deep learning (DL) aided multi-objective routing algorithm, which exploits the quasi-predictable network topology and operates in a distributed manner. Our simulation results based on real satellite, flight, and shipping data in the North Atlantic region show that the integrated network enhances the coverage quality by reducing the end-to-end (E2E) delay and by boosting the E2E throughput as well as improving the path-lifetime. The results demonstrate that our DL-aided multi-objective routing algorithm is capable of achieving near Pareto-optimal performance.