We report multispectral phase-shifting interference microscopy for quantitative phase imaging of human red blood cells (RBCs). A wide range of wavelengths are covered by means of using multiple color light emitting diodes (LEDs) with narrow spectral bandwidth ranging from violet to deep red color. The multicolor LED light source was designed and operated sequentially, which works as a multispectral scanning light source. Corresponding to each color LED source, five phase-shifted interferograms were recorded sequentially for the measurement of phase maps, as well as the refractive index of RBCs within the entire visible region. The proposed technique provides information about the effect of wavelengths on the morphology and refractive index of human RBCs. The system does not require expensive multiple color filters or any wavelength scanning mechanism along with broadband light source.