Mergers play a complex role in galaxy formation and evolution. Continuing to improve our understanding of these systems requires ever larger samples, which can be difficult (even impossible) to select from individual surveys. We use the new platform ESA Datalabs to assemble a catalog of interacting galaxies from the Hubble Space Telescope science archives; this catalog is larger than previously published catalogs by nearly an order of magnitude. In particular, we apply the Zoobot convolutional neural network directly to the entire public archive of HST F814W images and make probabilistic interaction predictions for 126 million sources from the Hubble Source Catalog. We employ a combination of automated visual representation and visual analysis to identify a clean sample of 21,926 interacting galaxy systems, mostly with z<1. Sixty-five percent of these systems have no previous references in either the NASA Extragalactic Database or Simbad. In the process of removing contamination, we also discover many other objects of interest, such as gravitational lenses, edge-on protoplanetary disks, and "backlit" overlapping galaxies. We briefly investigate the basic properties of this sample, and we make our catalog publicly available for use by the community. In addition to providing a new catalog of scientifically interesting objects imaged by HST, this work also demonstrates the power of the ESA Datalabs tool to facilitate substantial archival analysis without placing a high computational or storage burden on the end user.