The healthcare sector demands a higher degree of responsibility, trustworthiness, and accountability when implementing Artificial Intelligence (AI) systems. Machine learning operations (MLOps) for AI-based medical diagnostic systems are primarily focused on aspects such as data quality and confidentiality, bias reduction, model deployment, performance monitoring, and continuous improvement. However, so far, MLOps techniques do not take into account the need to provide resilience to disturbances such as adversarial attacks, including fault injections, and drift, including out-of-distribution. This article is concerned with the MLOps methodology that incorporates the steps necessary to increase the resilience of an AI-based medical diagnostic system against various kinds of disruptive influences. Post-hoc resilience optimization, post-hoc predictive uncertainty calibration, uncertainty monitoring, and graceful degradation are incorporated as additional stages in MLOps. To optimize the resilience of the AI based medical diagnostic system, additional components in the form of adapters and meta-adapters are utilized. These components are fine-tuned during meta-training based on the results of adaptation to synthetic disturbances. Furthermore, an additional model is introduced for post-hoc calibration of predictive uncertainty. This model is trained using both in-distribution and out-of-distribution data to refine predictive confidence during the inference mode. The structure of resilience-aware MLOps for medical diagnostic systems has been proposed. Experimentally confirmed increase of robustness and speed of adaptation for medical image recognition system during several intervals of the system's life cycle due to the use of resilience optimization and uncertainty calibration stages. The experiments were performed on the DermaMNIST dataset, BloodMNIST and PathMNIST. ResNet-18 as a representative of convolutional networks and MedViT-T as a representative of visual transformers are considered. It is worth noting that transformers exhibited lower resilience than convolutional networks, although this observation may be attributed to potential imperfections in the architecture of adapters and meta-adapters. The main novelty of the suggested resilience-aware MLOps methodology and structure lie in the separating possibilities and activities on creating a basic model for normal operating conditions and ensuring its resilience and trustworthiness. This is significant for the medical applications as the developer of the basic model should devote more time to comprehending medical field and the diagnostic task at hand, rather than specializing in system resilience. Resilience optimization increases robustness to disturbances and speed of adaptation. Calibrated confidences ensure the recognition of a portion of unabsorbed disturbances to mitigate their impact, thereby enhancing trustworthiness.