  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, gamma, to
parameterize the linear growth rate of cosmic structure. Using the method of
Mantz et al. (2009a), 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 systematic
uncertainties. We combine the cluster growth data with gas mass fraction, SNIa,
BAO and CMB data. This combination leads to a tight correlation between gamma
and sigma_8. Consistency with GR requires gamma~0.55. Under the assumption of
self-similar evolution and constant scatter in the scaling relations, and for a
flat LCDM model, we measure gamma(sigma_8/0.8)^6.8=0.55+0.13-0.10, with
0.79<sigma_8<0.89. 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 gamma that are only ~20% 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 LCDM. Our results represent the most robust
consistency test of GR on cosmological scales to date. (Abridged)
