How do societies make moral sense of generative AI during its public emergence? As large language models become embedded in everyday life, public discourse plays a critical role in shaping how these technologies are interpreted, contested, and governed. Drawing on Moral Foundations Theory, this study analyzes 2.7 million posts about generative AI on X (formerly Twitter) following the release of ChatGPT in late 2022. We examine how different social actors construct moral narratives around generative AI, how these narratives evolve over time, and whether they influence the visibility of AI-related discourse. Results show that moral language becomes less volatile over time, suggesting that initial moral contestation gives way to more stable interpretations of generative AI. Moral expression also differs systematically across actors: academic accounts employ more restrained and positive moral language, whereas news accounts frame generative AI more negatively. Finally, engagement analyses reveal a conditional moral virtue penalty, with positive moral language associated with lower sharing overall, while the relationship between moral intensity and engagement varies across elite actors. Together, these findings suggest that public discourse constitutes an important mechanism through which societies negotiate the meaning, legitimacy, and governance of generative AI, shaping which interpretations of emerging AI technologies gain prominence in online information ecosystems.