Data-driven medical AI is traditionally formulated as a discriminative mapping from input $X$ to output $Y$ via a learned function $f$, which does not generalize well across heterogeneous data and modalities encountered in real-world clinical settings. In this work, we propose a fundamentally different, generative paradigm. We model the joint distribution $P(X,Y)$ using diffusion models and reframe inference as a test-time output optimization problem. By guiding the generative process to match observed inputs, our framework enables flexible, gradient-based conditioning at inference time without architectural changes or retraining, effectively supporting arbitrary and previously unseen combinations of observations. Extensive experiments demonstrate strong performance across standard and cross-modality medical image segmentation, few-shot segmentation with only 2 or 4 training samples, degraded-input segmentation, shape completion from sparse and partial observations, and zero-shot application to demonstrate generality. To support these evaluations, we curated and released a large-scale text-shape dataset derived from MedShapeNet. Our results highlight the versatility of generative joint modeling as a foundation for reusable, task-agnostic medical AI systems.