Grammatical Error Correction (GEC) and grammatical acceptability judgment (COLA) are core tasks in natural language processing, sharing foundational grammatical knowledge yet typically evolving independently. This paper introduces COLA-GEC, a novel bidirectional framework that enhances both tasks through mutual knowledge transfer. First, we augment grammatical acceptability models using GEC datasets, significantly improving their performance across multiple languages. Second, we integrate grammatical acceptability signals into GEC model training via a dynamic loss function, effectively guiding corrections toward grammatically acceptable outputs. Our approach achieves state-of-the-art results on several multilingual benchmarks. Comprehensive error analysis highlights remaining challenges, particularly in punctuation error correction, providing insights for future improvements in grammatical modeling.