Primary aldosteronism (PA) is the most frequent cause of secondary hypertension. Early diagnoses of PA are essential to avoid the long-term negative effects of elevated aldosterone concentration on the cardiovascular and renal system. In this work, we study the texture of the carotid artery vessel wall from longitudinal ultrasound images in order to automatically distinguish between PA and essential hypertension (EH). The texture is characterized using 140 Haralick and 10 wavelet features evaluated in a region of interest in the vessel wall, followed by the XGBoost classifier. Carotid ultrasound studies were carried out on 33 patients aged 42-72 years with PA, 52 patients with EH, and 33 normotensive controls. For the most clinically relevant task of distinguishing PA and EH classes, we achieved a classification accuracy of 73% as assessed by a leave-one-out procedure. This result is promising even compared to the 57% prediction accuracy using clinical characteristics alone or 63% accuracy using a combination of clinical characteristics and intima-media thickness (IMT) parameters. If the accuracy is improved and the method incorporated into standard clinical procedures, this could eventually lead to an improvement in the early diagnosis of PA and consequently improve the clinical outcome for these patients in future.