Clinical decision-making relies on the integration of information across various data modalities, such as clinical time-series, medical images and textual reports. Compared to other domains, real-world medical data is heterogeneous in nature, limited in size, and sparse due to missing modalities. This significantly limits model performance in clinical prediction tasks. Inspired by clinical workflows, we introduce MedPatch, a multi-stage multimodal fusion architecture, which seamlessly integrates multiple modalities via confidence-guided patching. MedPatch comprises three main components: (i) a multi-stage fusion strategy that leverages joint and late fusion simultaneously, (ii) a missingness-aware module that handles sparse samples with missing modalities, (iii) a joint fusion module that clusters latent token patches based on calibrated unimodal token-level confidence. We evaluated MedPatch using real-world data consisting of clinical time-series data, chest X-ray images, radiology reports, and discharge notes extracted from the MIMIC-IV, MIMIC-CXR, and MIMIC-Notes datasets on two benchmark tasks, namely in-hospital mortality prediction and clinical condition classification. Compared to existing baselines, MedPatch achieves state-of-the-art performance. Our work highlights the effectiveness of confidence-guided multi-stage fusion in addressing the heterogeneity of multimodal data, and establishes new state-of-the-art benchmark results for clinical prediction tasks.