Longitudinal medical images are essential for monitoring disease progression by capturing spatiotemporal changes associated with dynamic biological processes. While current methods have made progress in modeling spatiotemporal patterns, they face three key limitations: (1) lack of generalizable framework applicable to diverse disease progression prediction tasks; (2) frequent overlook of the ordinal nature inherent in disease staging; (3) susceptibility to representation collapse due to structural similarities between adjacent time points, which can obscure subtle but discriminative progression biomarkers. To address these limitations, we propose a Generalizable LOngitudinal Medical Image Analysis framework for disease Progression prediction (GLOMIA-Pro). GLOMIA-Pro consists of two core components: progression representation extraction and progression-aware fusion. The progression representation extraction module introduces a piecewise orthogonal attention mechanism and employs a novel ordinal progression constraint to disentangle finegrained temporal imaging variations relevant to disease progression. The progression-aware fusion module incorporates a redesigned skip connection architecture which integrates the learned progression representation with current imaging representation, effectively mitigating representation collapse during cross-temporal fusion. Validated on two distinct clinical applications: knee osteoarthritis severity prediction and esophageal cancer treatment response assessment, GLOMIA-Pro consistently outperforms seven state-of-the-art longitudinal analysis methods. Ablation studies further confirm the contribution of individual components, demonstrating the robustness and generalizability of GLOMIA-Pro across diverse clinical scenarios.