Affective polarization and political sorting drive public antagonism around climate change and other issues at the science-policy nexus. We study cross-domain spillover of incivility in public engagements with climate change and public health on Twitter and Reddit using the COVID-19 period as a case study. We find strong evidence of the signatures of affective polarization surrounding COVID-19 spilling into the climate change domain. Across different social media systems, COVID-19 content is associated with incivility in climate discussions. These patterns of increased antagonism were responsive to pandemic events that made the link between science and public policy more salient. The observed spillover activated along pre-pandemic political cleavages, specifically anti-internationalist populist beliefs, that linked climate policy opposition to vaccine hesitancy. Our findings show how affective polarization in public engagement with science becomes entrenched across policy domains, which has implications for how the public engages with and communicates about issues such as climate change and public health.