A linear correlation has been proposed between the CO luminosity ($rm{L}^{prime}_{rm{CO}}$) and full-width at half maximum (FWHM) for high-redshift (z > 1) submillimeter galaxies. However, the controversy concerning the $rm{L}^{prime}_{rm{CO}}$-FWHM correlation seems to have been caused by the use of heterogeneous samples (e.g., different transition lines) and/or data with large measurement uncertainties. In order to avoid the uncertainty caused by using different rotational transitions, in this work we make an extensive effort to select only CO($J = 1-0$) data from the literature. We separate these wide-ranging redshift data into two samples : the low-redshift (z < 1) and high-redshift (z > 1) samples. The samples are corrected for lensing magnification factors if gravitational-lensing effects appeared in the observations. The correlation analysis shows that there exists significant $rm{L}^{prime}_{rm{CO}}$-FWHM correlations for both the low-redshift and high-redshift samples. A comparison of the low- and high-redshift $rm{L}^{prime}_{rm{CO}}$-FWHM correlations does not show strong evolution with redshift. Assuming that there is no evolution, we can use this relation to determine the model independent distances of high-redshift galaxies. We then constrain cosmological models with the calibrated high-redshift CO data and the sample of Type Ia supernovae in the Union 2.1 compilation. In the constraint for wCDM with our samples, the derived values are w_{0} = -1.02 {pm} 0.17, {Omega}_{m0} = 0.30{pm}0.02, and H_{0} = 70.00 {pm}0.60 km,s^{-1},Mpc^{-1}.
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