Context. The Mid-Infrared Instrument (MIRI) onboard the James Webb Space Telescope (JWST) will provide imaging, coronagraphy, low-resolution spectroscopy, and medium-resolution spectroscopy at unprecedented sensitivity levels in the mid-infrared wavelength range. The Medium Resolution Spectrometer (MRS) of MIRI is an integral field spectrograph that provides diffraction-limited spectroscopy between 4.9 and 28.3 μm, within a field of view (FOV) varying from ∼13 to ∼56 arcsec square. The design for MIRI MRS conforms with the goals of the JWST mission to observe high redshift galaxies and to study cosmology as well as observations of galactic objects, and stellar and planetary systems. Aims. From ground testing, we calculate the physical parameters essential for general observers and calibrating the wavelength solution and resolving power of the MRS which is critical for maximizing the scientific performance of the instrument. Methods. We have used ground-based observations of discrete spectral features in combination with Fabry-Perot etalon spectra to characterize the wavelength solution and spectral resolving power of the MRS. We present the methodology used to derive the MRS spectral characterization, which includes the precise wavelength coverage of each MRS sub-band, computation of the resolving power as a function of wavelength, and measuring slice-dependent spectral distortions. Results. The ground calibration of the MRS shows that it will cover the wavelength ranges from 4.9 to 28.3 μm, divided in 12 overlapping spectral sub-bands. The resolving power is R 3500 in channel 1, R 3000 in channel 2, R 2500 in channel 3, and R 1500 in channel 4. The MRS spectral resolution optimizes the sensitivity for detection of spectral features with a velocity width of ∼100 km s-1 which is characteristic of most astronomical phenomena JWST aims to study in the mid-infrared. Based on the ground test data, the wavelength calibration accuracy is estimated to be below one-tenth of a pixel (0.1 nm at 5 μm and 0.4 at 28 μm), with small systematic shifts due to the target position within a slice for unresolved sources that have a maximum amplitude of about 0.25 spectral resolution elements. The absolute wavelength calibration is presently uncertain at the level of 0.35 nm at 5 μm and 46 nm at 28 μm, and it will be refined using in-flight commissioning observations. Conclusions. Based on ground test data, the MRS complies with the spectral requirements for both the R and wavelength accuracy for which it was designed. We also present the commissioning strategies and targets that will be followed to update the spectral characterization of the MRS.