Underwater videos are vital for marine science, yet often suffer from color distortion, blurring, and low resolution due to environmental complexity and hardware constraints. The scarcity of paired high-quality data further limits the applicability of supervised enhancement approaches. In this work, we present SSE-UVSR, a novel self-supervised end-to-end underwater video super-resolution framework which, unlike conventional blind VSR methods, explicitly integrates underwater physical degradation modeling into the reconstruction process. Our method first estimates clean video content, transmission maps, and atmospheric light via a Degradation-Aware Module (DAM). An Edge Enhancement and Detail Preservation Module (DRM) then restores fine textures and sharp structures by leveraging underwater-specific priors. The entire model is trained without high-quality references, employing a cycle consistency loss together with re-degradation validation, which ensures both perceptual quality and physical plausibility. Experiments on synthetic and real-world underwater datasets across diverse conditions demonstrate that SSE-UVSR consistently outperforms existing methods in terms of both resolution and visual quality.