Precise measurement of micro-dispersion for optical devices (optical fiber, lenses, etc.) holds paramount significance across domains such as optical fiber communication and dispersion interference ranging. However, due to its complex system, complicated process, and low reliability, the traditional dispersion measurement methods (interference, phase shift, or time delay methods) are not suitable for the accurate measurement of micro-dispersion in a wide spectral range. Here, we propose a spectral-interferometry-based diff-iteration (SiDi) method for achieving accurate wide-band micro-dispersion measurements. Using an optical frequency comb, based on the phase demodulation of the dispersion interference spectrum, we employ the carefully designed SiDi method to solve the dispersion curve at any position and any order. Our approach is proficient in precisely measuring micro-dispersion across a broadband spectrum, without the need for cumbersome wavelength scanning processes or reliance on complex high-repetition-rate combs, while enabling adjustable resolution. The efficacy of the proposed method is validated through simulations and experiments. We employed a chip-scaled soliton microcomb (SMC) to compute the dispersion curves of a 14 m single-mode fiber (SMF) and a 0.05 m glass. Compared to a laser interferometer or the theoretical value given by manufacturers, the average relative error of refractive index measurement for single-mode fiber (SMF) reaches $2.8\times {10}^{-6}$ and for glass reaches $3.8\times {10}^{-6}$. The approach ensures high precision, while maintaining a simple system structure, with realizing adjustable resolution, thereby propelling the practical implementation of precise measurement and control-dispersion.