Clinical time series data from electronic health records and medical registries offer unprecedented opportunities to understand patient trajectories and inform medical decision-making. However, leveraging such data presents significant challenges due to irregular sampling, complex latent physiology, and inherent uncertainties in both measurements and disease progression. To address these challenges, we propose a generative modeling framework based on latent neural stochastic differential equations (SDEs) that views clinical time series as discrete-time partial observations of an underlying controlled stochastic dynamical system. Our approach models latent dynamics via neural SDEs with modality-dependent emission models, while performing state estimation and parameter learning through variational inference. This formulation naturally handles irregularly sampled observations, learns complex non-linear interactions, and captures the stochasticity of disease progression and measurement noise within a unified scalable probabilistic framework. We validate the framework on two complementary tasks: (i) individual treatment effect estimation using a simulated pharmacokinetic-pharmacodynamic (PKPD) model of lung cancer, and (ii) probabilistic forecasting of physiological signals using real-world intensive care unit (ICU) data from 12,000 patients. Results show that our framework outperforms ordinary differential equation and long short-term memory baseline models in accuracy and uncertainty estimation. These results highlight its potential for enabling precise, uncertainty-aware predictions to support clinical decision-making.