Distribution shift is a common challenge in medical images obtained from different clinical centers, significantly hindering the deployment of pre-trained semantic segmentation models in real-world applications across multiple domains. Continual Test-Time Adaptation(CTTA) has emerged as a promising approach to address cross-domain shifts during continually evolving target domains. Most existing CTTA methods rely on incrementally updating model parameters, which inevitably suffer from error accumulation and catastrophic forgetting, especially in long-term adaptation. Recent prompt-tuning-based works have shown potential to mitigate the two issues above by updating only visual prompts. While these approaches have demonstrated promising performance, several limitations remain:1)lacking multi-scale prompt diversity, 2)inadequate incorporation of instance-specific knowledge, and 3)risk of privacy leakage. To overcome these limitations, we propose Multi-scale Global-Instance Prompt Tuning(MGIPT), to enhance scale diversity of prompts and capture both global- and instance-level knowledge for robust CTTA. Specifically, MGIPT consists of an Adaptive-scale Instance Prompt(AIP) and a Multi-scale Global-level Prompt(MGP). AIP dynamically learns lightweight and instance-specific prompts to mitigate error accumulation with adaptive optimal-scale selection mechanism. MGP captures domain-level knowledge across different scales to ensure robust adaptation with anti-forgetting capabilities. These complementary components are combined through a weighted ensemble approach, enabling effective dual-level adaptation that integrates both global and local information. Extensive experiments on medical image segmentation benchmarks demonstrate that our MGIPT outperforms state-of-the-art methods, achieving robust adaptation across continually changing target domains.