Semen quality assessed by sperm count and sperm cell characteristics such as morphology and motility, is considered to be the main determinant of men's reproductive health. Therefore, sperm cell selection is vital in assisted reproductive technology (ART) used for the treatment of infertility. Conventional bright field optical microscopy is widely utilized for the imaging and selection of sperm cells based on the qualitative analysis by experienced clinicians. In this study, we report the development of a highly sensitive quantitative phase microscopy (QPM) using partially spatially coherent light source, which is a label-free, non-invasive and high-resolution technique to quantify various biophysical parameters. The partial spatial coherence nature of light source provides a significant improvement in spatial phase sensitivity and hence reconstruction of the phase of the entire sperm cell is demonstrated, which was otherwise not possible using highly spatially coherent light source. High sensitivity of the system enables quantitative phase imaging of the specimens having very low refractive index contrast with respect to the medium like tail of the sperm cells. Further, it also benefits with accurate quantification of 3D-morphological parameters of sperm cells which might be helpful in the infertility treatment. The quantitative analysis of more than 2500 sperm cells under hydrogen peroxide (H2O2) induced oxidative stress condition is demonstrated. It is further correlated with motility of sperm cell to study the effect of oxidative stress on healthy sperm cells. The results exhibit a decrease in the maximum phase values of the sperm head as well as decrease in the sperm cell's motility with increasing oxidative stress, i.e., H2O2 concentration. Various morphological and texture parameters were extracted from the phase maps and subsequently support vector machine (SVM) based machine learning algorithm is employed for the classification of the control and the stressed sperms cells. The algorithm achieves an area under the receiver operator characteristic (ROC) curve of 89.93% based on the all morphological and texture parameters with a sensitivity of 91.18%. The proposed approach can be implemented for live sperm cells selection in ART procedure for the treatment of infertility.