Rejecting cosmic rays (CRs) is essential for scientific interpretation of CCD-captured data, but detecting CRs in single-exposure images has remained challenging. Conventional CR-detection algorithms require tuning multiple parameters experimentally making it hard to automate across different instruments or observation requests. Recent work using deep learning to train CR-detection models has demonstrated promising results. However, instrument-specific models suffer from performance loss on images from ground-based facilities not included in the training data. In this work, we present Cosmic-CoNN, a deep-learning framework designed to produce generic CR-detection models. We build a large, diverse ground-based CR dataset leveraging thousands of images from the Las Cumbres Observatory global telescope network to produce a generic CR-detection model which achieves a 99.91% true-positive detection rate and maintains over 96.40% true-positive rates on unseen data from Gemini GMOS-N/S, with a false-positive rate of 0.01%. Apart from the open-source framework and dataset, we also build a suite of tools including console commands, a web-based application, and Python APIs to make automatic, robust CR detection widely accessible by the community of astronomers.