The spread of misinformation, especially health-related misinformation has raised concerns globally. As an immediate remedy, fact-checking has been identified as an important solution. Adopting a 2 (source credibility: high vs. low) × 2 (source consistency: consistent vs. inconsistent) × 3 (ways of correction: human fact-checking vs. AI fact-checking vs. simple rebuttal) factorial design experiment (N = 754), this study examined how ways of correction and source consistency may affect individuals' intentions to share health misinformation correction on social media on two health topics: sunscreen safety and vaccine safety. Results showed that human and AI fact-checking correction elicited higher sharing intention compared to simple rebuttal. Correction coming from a different source than the original misinformation source elicited higher sharing intention, compared to correction from the same source. Perceived correction source credibility mediated the effects of ways of correction and source consistency on correction sharing intention. Theoretical and practical implications were discussed.