The peculiar velocity distribution of cluster member galaxies provides a powerful tool to directly investigate the gravitational potentials within galaxy clusters and to test the gravity theory on megaparsec scales. We exploit spectroscopic galaxy and galaxy cluster samples extracted from the latest releases of the Sloan Digital Sky Survey (SDSS) to derive new constraints on the gravity theory. We consider a spectroscopic sample of $3058$ galaxy clusters, with a maximum redshift of $0.5$. We analyse the velocity distribution of the cluster member galaxies to make new measurements of the gravitational redshift effect inside galaxy clusters. We accurately estimate the cluster centres, computing them as the average of angular positions and redshifts of the closest galaxies to the brightest cluster galaxies. We find that this centre definition provides a better estimation of the centre of the cluster gravitational potential wells, relative to simply assuming the brightest cluster galaxies as the cluster centres, as done in the past literature. We compare our measurements with the theoretical predictions of three different gravity theories: general relativity (GR), the $f(R)$ model, and the Dvali-Gabadadze-Porrati (DGP) model. A new statistical procedure is used to fit the measured gravitational redshift signal and thus to discriminate among the considered gravity theories. Finally, we investigate the systematic uncertainties possibly affecting the analysis. We clearly detect the gravitational redshift effect in the exploited cluster member catalogue. We recover an integrated gravitational redshift signal of $-11.4 \pm 3.3$ km s$^{-1}$, which is in agreement, within the errors, with past literature works. Overall, our results are consistent with both GR and DGP predictions, while they are in marginal disagreement with the predictions of the considered $f(R)$ strong field model.