We study the predictions for structure formation in an induced gravity dark energy model with a quartic potential. By developing a dedicated Einstein-Boltzmann code, we study self-consistently the dynamics of homogeneous cosmology and of linear perturbations without using any parametrization. By evolving linear perturbations with initial conditions in the radiation era, we accurately recover the quasi-static analytic approximation in the matter dominated era. We use PLANCK 2013 data and a compilation of baryonic acoustic oscillation (BAO) data to constrain the coupling γ to the Ricci curvature and the other cosmological parameters. By connecting the gravitational constant in the Einstein equation to the one measured in a Cavendish-like experiment, we find γ < 0.0012 at 95% CL with PLANCK 2013 and BAO data. This is the tightest cosmological constraint on γ and on the corresponding derived post-Newtonian parameters. Because of a degeneracy between γ and the Hubble constant H0, we show how larger values for γ are allowed, but not preferred at a significant statistical level, when local measurements of H0 are combined in the analysis with PLANCK 2013 data.