Residential greenness has been associated with reduced heat-related illness, yet the specific role of greenspace morphology at the neighborhood scale remains insufficiently understood. This study quantified the relationship between heat-related illness and multiple dimensions of greenspace morphology using an eight year (2016-2023) unbalanced panel dataset comprising 19,021 block group year observations across 2,427 census block groups in Arizona, USA. One meter high resolution National Agricultural Imagery Program aerial imagery was classified to calculate greenspace percentage, number of greenspaces, average size, shape complexity, connectedness, and distantness, at the block group level. We applied conditional spatial autoregressive models with a negative binomial distribution to estimate associations between each morphology metric and yearly heat-related illness counts, adjusting for sociodemographic and geographic covariates. We found higher greenspace percentage, aggregation, shape complexity, connectedness, and density were consistently associated with lower heat-related illness risk. A one standard deviation increases in shape complexity corresponded to a 12.4% decrease in expected heat-related illness counts (IRR=0.876, 95% CI: 0.834-0.921). Similarly, increases in greenspace percentage (14.6% decrease; IRR=0.855, 95% CI: 0.827-0.885), number of greenspace patches (3.7% decrease; IRR=0.963, 95% CI: 0.937-0.990), average size (4.5% decrease; IRR=0.955, 95% CI: 0.923-0.989), and connectedness (5.5% decrease; IRR=0.945, 95% CI: 0.918-0.972) were all protective. In contrast, larger inter greenspace distances were associated with increased heat-related illness risk (6.1% increase; IRR=1.061, 95% CI: 1.033-1.091). Our findings highlight the critical importance of multiple dimensions of greenspace morphology in mitigating heat-related health risks. These results suggest that heat reduction planning with greening initiatives should consider not only the amount of greenspace but also its spatial configuration to maximize cooling and result in health benefits.