Missing or corrupted modalities are common in physiological signal-based medical applications owing to hardware constraints or motion artifacts. However, most existing methods assume the availability of all modalities, resulting in substantial performance degradation in the absence of any modality. To overcome this limitation, this study proposes PhysioME, a robust framework designed to ensure reliable performance under missing modality conditions. PhysioME adopts: (1) a multimodal self-supervised learning approach that combines contrastive learning with masked prediction; (2) a Dual-PathNeuroNet backbone tailored to capture the temporal dynamics of each physiological signal modality; and (3) a restoration decoder that reconstructs missing modality tokens, enabling flexible processing of incomplete inputs. The experimental results show that PhysioME achieves high consistency and generalization performance across various missing modality scenarios. These findings highlight the potential of PhysioME as a reliable tool for supporting clinical decision-making in real-world settings with imperfect data availability.