Rapid appraisal of damages related to hazard events is important to first responders, government agencies, insurance industries, and other private and public organizations. While satellite monitoring, ground-based sensor systems, inspections, and other technologies provide data to inform post-disaster response, crowdsourcing through social media is an additional and novel data source. In this study, the use of social media data, principally Twitter postings, is investigated to make approximate but rapid early assessments of damages following earthquake disasters. The goal is to explore the potential utility of using social media data for rapid damage assessment after sudden-onset hazard events and to identify insights related to potential challenges. This study defines a text-based damage assessment scale for earthquake damages and then develops a text classification model for rapid damage assessment. The 2019 Ridgecrest, California earthquake sequence is mainly investigated as the case study. Results reveal that Twitter users rapidly responded to this sudden-onset event, and the damage estimation shows temporal and spatial characteristics. The generalization ability of the model is validated through the investigation of damage assessment for another five earthquake events. Although the accuracy remains a challenge compared to ground-based instrumental readings and inspections, the proposed damage assessment model features rapidity with large amounts of data at spatial densities that exceed those of conventional sensor networks.