Blood vessel segmentation is a core task in medical image analysis for the care of vascular diseases and surgical planning, yet the challenges of providing expert vascular annotations pose a major obstacle for the progress of related deep learning techniques. To address this, we propose VesselSim, a two-stage framework for universal 3D blood vessel segmentation that eliminates the need for real annotated data during training. First, we introduce a stochastic, geometry-driven vascular simulation framework that models recursive branching, curvature-controlled growth, and collision-aware topology, followed by domain-randomized intensity synthesis to generate 16,500 anatomically plausible 3D angiographic volumes. Second, a 3D U-Net is trained solely on this synthetic data. To bridge the domain gap from synthetic to real images at inference time, we introduce a test-time adaptation strategy via a self-supervised mask reconstruction decoder, enabling adaptation to unseen clinical scans without prior domain knowledge. We evaluate VesselSim in a zero-shot setting on multiple real-world datasets spanning MR and CT across several anatomical regions, including the brain and kidneys. Despite being trained exclusively on synthetic data, VesselSim achieves performance competitive with state-of-the-art vascular segmentation foundation models. These findings suggest that learning vessel geometry from synthetic tubular structures is effective for robust cross-domain generalization, substantially reducing the reliance on acquired medical imaging data and more importantly, expert annotations.