Drought is one of the major natural disasters in the world, and it is important to understand and grasp the spatiotemporal evolution characteristics of drought for drought monitoring and prevention. In this study, the standardized precipitation evapotranspiration index (SPEI) was calculated based on monthly precipitation and average temperature data from 15 meteorological stations in Henan Province, China during the period 1987–2020. The spatiotemporal evolution characteristics of droughts in Henan Province were analyzed using run theory and Moran's I, and drought prediction was carried out using grey disaster theory. The results show that: (1) During the period 1987–2020, the overall drought in Henan Province shows a slow intensification trend, and SPEI-12 has a certain adaptability in the drought characterization of Henan Province; (2) The frequency of droughts in the northern and eastern regions of Henan Province is high, the southern region is prone to continuous drought, the drought ephemeris and intensity show a strong positive correlation, and the spatial spillover effect of autumn drought is the most obvious; (3) The grey disaster prediction model has excellent accuracy and predicts that the next possible drought in Henan Province will be in the east of Henan Province in the spring of 2023.