Green space equity is increasingly recognized as a critical environmental condition for healthy aging, yet existing research often overlooks how different green space attributes—accessibility and diversity—are associated with distinct dimensions of older adults’ health. Limited attention has been paid to their nonlinear threshold effects or to the social pathways through which green spaces influence health outcomes. Using the United States county-level panel data from 2020 to 2023, this study integrates fixed-effects models, Extreme Gradient Boosting (XGBoost), and mediation analysis to examine the associations between green accessibility measured by the Two-Step Floating Catchment Area (2SFCA) method, and green diversity measured by the Shannon Index, on the general, physical, and mental health of older adults. Findings indicate that (1) higher green accessibility is associated with better general health, whereas green diversity shows a stronger association with physical health, reflecting its link to more heterogeneous ecosystem service environments. (2) Green accessibility demonstrates the threshold effect, in which the strength of association with health becomes steeper once accessibility approaches higher levels. (3) Green space equity is linked to health partly through social structures. Education clustering and marital stability mediate the associations with general health, while mental health appears to depend more on the social interaction opportunities embedded within green environments than on their physical attributes alone. The study proposes an integrated “physical environment–social structure–health outcome” framework and a threshold-oriented spatial intervention strategy, highlighting the need to prioritize improvements in green accessibility in underserved areas and prioritizing green diversity and age-friendly social functions where accessibility is already high. These findings offer evidence for designing inclusive, health-oriented urban environments for aging populations.