Depth perception plays an essential role in the viewer experience for immersive virtual reality (VR) visual environments. However, previous research investigations in the depth quality of 3D/stereoscopic images are rather limited, and in particular, are largely lacking for 3D viewing of 360-degree omnidirectional content. In this work, we make one of the first attempts to develop an objective quality assessment model named depth quality index (DQI) for efficient no-reference (NR) depth quality assessment of stereoscopic omnidirectional images. Motivated by the perceptual characteristics of the human visual system (HVS), the proposed DQI is built upon multi-color-channel, adaptive viewport selection, and interocular discrepancy features. Experimental results demonstrate that the proposed method outperforms state-of-the-art image quality assessment (IQA) and depth quality assessment (DQA) approaches in predicting the perceptual depth quality when tested using both single-viewport and omnidirectional stereoscopic image databases. Furthermore, we demonstrate that combining the proposed depth quality model with existing IQA methods significantly boosts the performance in predicting the overall quality of 3D omnidirectional images.