This is the third of a series of papers in which we derive simultaneous constraints on cosmological parameters and X-ray scaling relations using observations of the growth of massive, X-ray flux-selected galaxy clusters. Our data set consists of 238 clusters drawn from the ROSAT All-Sky Survey and incorporates extensive follow-up observations using the Chandra X-ray Observatory. Here we present improved constraints on departures from general relativity (GR) on cosmological scales, using the growth index, γ, to parametrize the linear growth rate of cosmic structure. Using the method of Mantz et al. (Paper I), we simultaneously and self-consistently model the growth of X-ray luminous clusters and their observable–mass scaling relations, accounting for survey biases, parameter degeneracies and the impact of systematic uncertainties. Such analysis of the survey and follow-up data is crucial, else spurious constraints may be obtained. We combine the X-ray cluster growth data with cluster gas mass fraction, Type Ia supernova, baryon acoustic oscillation and cosmic microwave background data. We find that the combination of these data leads to a tight correlation between γ and the normalization of the matter power spectrum, σ8. Consistency with GR requires a measured growth index of γ∼ 0.55. Under the assumption of self-similar evolution and constant scatter in the cluster observable–mass scaling relations, and for a spatially flat model with a cosmological constant, we measure γ(σ8/0.8)6.8= 0.55+0.13−0.10, with allowed values for σ8 in the range of 0.79–0.89 (68.3 per cent confidence limits). Relaxing the assumptions on the scaling relations by introducing two additional parameters to model possible evolution in the normalization and scatter of the luminosity–mass relation, we obtain consistent constraints on γ that are only ∼20 per cent weaker than those above. Allowing the dark energy equation of state, w, to take any constant value, we simultaneously constrain the growth and expansion histories and find no evidence for departures from either GR or the cosmological constant plus cold dark matter paradigm. Our results represent the most robust consistency test of GR on cosmological scales to date.