Rejecting cosmic rays (CRs) is essential for scientific interpretation of CCD-captured data, but detecting CRs in single-exposure images has remained challenging. Conventional algorithms require experimental parameter tuning for different instruments, recent work using deep learning produces instrument-specific models that suffer from performance loss on new facilities not included in the training data. In this work, we present Cosmic-CoNN, a generic CR detector deployed at the Las Cumbres Observatory (LCO) for over 20 telescopes. We first leverage thousands of images from LCO's global telescope network to build a large, diverse ground-based CR dataset for rich coverage of instruments and CR features. We then use this new dataset, our deep-learning framework, and a novel Median-Weighted loss function designed for CR-detection to train a generic model that achieves a 99.91% true-positive detection rate on LCO data and maintains over 96.40% on unseen data from Gemini GMOS-N/S, with a false-positive rate of 0.01%. We also build a suite of tools including an interactive CR mask visualization and editing interface, console commands, and Python APIs to make automatic, robust CR detection widely accessible by the community of astronomers. Our open-source dataset, codebase, and trained models are available at https://github.com/cy-xu/cosmic-conn.