Modern advances in machine learning (ML) and wearable medical sensors (WMSs) in edge devices have enabled ML-driven disease detection for smart healthcare. Conventional ML-driven methods for disease detection rely on customizing individual models for each disease and its corresponding WMS data. However, such methods lack adaptability to distribution shifts and new task classification classes. In addition, they need to be rearchitected and retrained from scratch for each new disease. Moreover, installing multiple ML models in an edge device consumes excessive memory, drains the battery faster, and complicates the detection process. To address these challenges, we propose DOCTOR, a multi-disease detection continual learning (CL) framework based on WMSs. It employs a multi-headed deep neural network (DNN) and a replay-style CL algorithm. The CL algorithm enables the framework to continually learn new missions where different data distributions, classification classes, and disease detection tasks are introduced sequentially. It counteracts catastrophic forgetting with a data preservation method and a synthetic data generation (SDG) module. The data preservation method preserves the most informative subset of real training data from previous missions for exemplar replay. The SDG module models the probability distribution of the real training data and generates synthetic data for generative replay while retaining data privacy. The multi-headed DNN enables DOCTOR to detect multiple diseases simultaneously based on user WMS data. We demonstrate DOCTOR's efficacy in maintaining high disease classification accuracy with a single DNN model in various CL experiments. In complex scenarios, DOCTOR achieves 1.43 times better average test accuracy, 1.25 times better F1-score, and 0.41 higher backward transfer than the naive fine-tuning framework with a small model size of less than 350KB.