In this paper, we study the normalised characteristic scale of transition to cosmic homogeneity, $\mathcal{R}_H/d_V$, as a cosmological probe. We use a compilation of SDSS galaxy samples, comprising more than $10^6$ galaxies in the redshift range $0.17 \leq z \leq 2.2$ within the largest comoving volume to date, $\sim 8 h^{-3}\mathrm{Gpc}^3$. We show that these samples can be described by a single bias model as a function of redshift. By combining our measurements with prior Cosmic Microwave Background and Lensing information from the Planck satellite, we constrain the total matter density ratio of the universe, $\Omega_m = 0.363 \pm 0.025$, and the Dark Energy density ratio, $\Omega_{\Lambda} = 0.649 \pm 0.021$, improving the values from Planck alone by 31% and 28%, respectively. Our results are compatible with a flat $\Lambda$CDM model. These results show the complementarity of the normalised homogeneity scale with other cosmological probes and open new roads to cosmometry.