Neural architectures have emerged in recent years as potentially enhanced solutions to the longstanding underwater acoustic localization problem. In this unique domain, nontrivial physical phenomena, such as depth-varying speed of sound, play a key role in the environment-dependent propagation model. Consequently, extracting location-related information of acoustic emitters, encapsulated in the relevant channel response, is both analytically and computationally challenging. Thus, deep neural networks (DNNs), which can circumvent the need for exact analytical characterizations and solutions, have been considered a prospective alternative to classical approaches. However, localization systems are typically required to be robust (in several respects), a property that DNNs do not necessarily possess. In this work, we focus on this critical aspect and show through a diverse set of simulations that our DNN-based localizer consistently manifests such robustness. Specifically, we focus on out-of-distribution generalization for input data, and show that our model remains performant under various distributional deviations.