The self-trapped exciton magnetic polaron (ST-EXMP) in diluted magnetic semiconductors (DMS's) has been the subject of intensive investigations related to the observed $L2$ photoluminescence during the past two decades, although the stability seems still in controversy. In this paper we have studied the stability condition for ST-EXMP theoretically: we applied the molecular-field approximation to begin with, and then incorporated the effects of the thermodynamic fluctuations of magnetization. The calculation suggests that the exciton magnetic polaron (EXMP) in ${\mathrm{Cd}}_{1\ensuremath{-}x}{\mathrm{Mn}}_{x}\mathrm{Te}$ $(x=0.05--0.2)$ is not self-trapped except at quite low temperatures below \ensuremath{\sim}1 K, but the other primary localization mechanisms, for example, alloy potential fluctuations, are necessary for the localization at the temperatures above \ensuremath{\sim}1 K. This conclusion is different from a recent theoretical report [J. Appl. Phys. 81, 6297 (1997)] that the ST-EXMP in DMS's is stable up to relatively high temperatures $(T<~30\mathrm{K})$ over a wide range of Mn concentrations without any other primary localization. The reason for this difference is clarified. It is also discussed briefly that when the higher-order sp-d exchange interaction is taken into account, the ST-EXMP in DMS's may become more unstable than that studied in the present paper.