Universal medical image segmentation seeks to use a single foundational model to handle diverse tasks across multiple imaging modalities. However, existing approaches often rely heavily on manual visual prompts or retrieved reference images, which limits their automation and robustness. In addition, naive joint training across modalities often fails to address large domain shifts. To address these limitations, we propose Concept-to-Pixel (C2P), a novel prompt-free universal segmentation framework. C2P explicitly separates anatomical knowledge into two components: Geometric and Semantic representations. It leverages Multimodal Large Language Models (MLLMs) to distill abstract, high-level medical concepts into learnable Semantic Tokens and introduces explicitly supervised Geometric Tokens to enforce universal physical and structural constraints. These disentangled tokens interact deeply with image features to generate input-specific dynamic kernels for precise mask prediction. Furthermore, we introduce a Geometry-Aware Inference Consensus mechanism, which utilizes the model's predicted geometric constraints to assess prediction reliability and suppress outliers. Extensive experiments and analysis on a unified benchmark comprising eight diverse datasets across seven modalities demonstrate the significant superiority of our jointly trained approach, compared to universe- or single-model approaches. Remarkably, our unified model demonstrates strong generalization, achieving impressive results not only on zero-shot tasks involving unseen cases but also in cross-modal transfers across similar tasks. Code is available at: https://github.com/Yundi218/Concept-to-Pixel