  We discuss the ability of the planned Euclid mission to detect deviations
from General Relativity using its extensive redshift survey of more than 50
Million galaxies. Constraints on the gravity theory are placed measuring the
growth rate of structure within 14 redshift bins between z=0.7 and z=2. The
growth rate is measured from redshift-space distortions, i.e. the anisotropy of
the clustering pattern induced by coherent peculiar motions. This is performed
in the overall context of the Euclid spectroscopic survey, which will
simultaneously measure the expansion history of the universe, using the power
spectrum and its baryonic features as a standard ruler, accounting for the
relative degeneracies of expansion and growth parameters. The resulting
expected errors on the growth rate in the different redshift bins, expressed
through the quantity f\sigma_8, range between 1.3% and 4.4%. We discuss the
optimisation of the survey configuration and investigate the important
dependence on the growth parameterisation and the assumed cosmological model.
We show how a specific parameterisation could actually drive the design towards
artificially restricted regions of the parameter space. Finally, in the
framework of the popular "\gamma -parameterisation", we show that the Euclid
spectroscopic survey alone will already be able to provide substantial evidence
(in Bayesian terms) if the growth index differs from the GR value \gamma=0.55
by at least \sim 0.13. This will combine with the comparable inference power
provided by the Euclid weak lensing survey, resulting in Euclid's unique
ability to provide a decisive test of modified gravity.
