Modern vision-language models (VLMs) develop patch embedding and convolution backbone within vector space, especially Euclidean ones, at the very founding. When expanding VLMs to a galaxy-scale for understanding astronomical phenomena, the integration of spherical space for planetary orbits and hyperbolic spaces for black holes raises two formidable challenges. a) The current pre-training model is confined to Euclidean space rather than a comprehensive geometric embedding. b) The predominant architecture lacks suitable backbones for anisotropic physical geometries. In this paper, we introduced Galaxy-Walker, a geometry-aware VLM, for the universe-level vision understanding tasks. We proposed the geometry prompt that generates geometry tokens by random walks across diverse spaces on a multi-scale physical graph, along with a geometry adapter that compresses and reshapes the space anisotropy in a mixture-of-experts manner. Extensive experiments demonstrate the effectiveness of our approach, with Galaxy-Walker achieving state-of-the-art performance in both galaxy property estimation (R² scores up to 0.91) and morphology classification tasks (up to +0.17 F1 improvement in challenging features), significantly outperforming both domain-specific models and general-purpose VLMs.