Automatically detecting stress in speech provides an unobtrusive way to gain insights relevant to behavioral research or clinical assessment. This study investigates the automatic differentiation between a stressful and non-stressful situation, and the prediction of physiological and affective stress responses. Speech data was collected from 50 participants who either completed the Trier Social Stress Test (TSST) or a non-stressful control condition. With a processing pipeline that included speaker diarization and machine learning models, we achieved stress detection performance significantly above a mean baseline. Moreover, relevant physiological and affective stress responses were partially predictable from acoustic-prosodic features. Feature-importance analyses identified the most informative predictors contributing to model performance. The findings demonstrate that speech can serve as a meaningful and unobtrusive indicator of multiple dimensions of the human stress response.