We demonstrate the use of a low-coherence spectral-domain phase microscopy (SDPM) system for accurate quantitative phase measurements in red blood cells (RBCs) for the prognosis and monitoring of disease conditions that affect the visco-elastic properties of RBCs. Using the system, we performed time-recordings of cell membrane fluctuations, and compared the nano-scale fluctuation dynamics of healthy and glutaraldehyde-treated RBCs. Glutaraldehyde-treated RBCs possess a lower amplitude of fluctuations reflecting an increased membrane stiffness. To demonstrate the ability of our system to measure fluctuations of lower amplitudes than those measured by the commonly used holographic phase microscopy techniques, we also constructed a wide-field digital interferometric microscope and compared the performances of the two systems. Due to its common-path geometry, the optical-path-delay stability of SDPM was found to be less than 0.3nm in liquid environment, at least three times better than in holographic phase microscopy under the same conditions. In addition, due to the compactness of SDPM and its inexpensive and robust design, the system possesses a high potential for clinical applications.