Human gene interaction networks, commonly known as interactomes, encode genes functional relationships, which are invaluable knowledge for translational medical research and the mechanistic understanding of complex human diseases. Meanwhile, the advancement of network embedding techniques has inspired recent efforts to identify novel human disease-associated genes using canonical interactome embeddings. However, one pivotal challenge that persists stems from the fact that many complex diseases manifest in specific biological contexts, such as tissues or cell types, and many existing interactomes do not encapsulate such information. Here, we propose CONE, a versatile approach to generate context-specific embeddings from a context-free interactome. The core component of CONE consists of a graph attention network with contextual conditioning, and it is trained in a noise contrastive fashion using contextualized interactome random walks localized around contextual genes. We demonstrate the strong performance of CONE embeddings in identifying disease-associated genes when using known associated biological contexts to the diseases. Furthermore, our approach offers insights into understanding the biological contexts associated with human diseases.