Aims: The role of galaxy mergers in massive galaxy evolution, and in particular to mass assembly and size growth, remains an open question. In this paper we measure the merger fraction and rate, both minor and major, of massive early-type galaxies (M ⋆ ≥ 1011 M&sun;) in the COSMOS field, and study their role in mass and size evolution. Methods: We used the 30-band photometric catalogue in COSMOS, complemented with the spectroscopy of the zCOSMOS survey, to define close pairs with a separation on the sky plane 10 h-1 kpc ≤ rp ≤ 30 h-1 kpc and a relative velocity Δv ≤ 500 km s-1 in redshift space. We measured both major (stellar mass ratio μ ≡ M ⋆ ,2/M ⋆ ,1 ≥ 1/4) and minor (1/10 ≤ μ < 1/4) merger fractions of massive galaxies, and studied their dependence on redshift and on morphology (early types vs. late types). Results: The merger fraction and rate of massive galaxies evolves as a power-law (1 + z)n, with major mergers increasing with redshift, nMM = 1.4, and minor mergers showing little evolution, nmm ~ 0. When split by their morphology, the minor merger fraction for early-type galaxies (ETGs) is higher by a factor of three than that for late-type galaxies (LTGs), and both are nearly constant with redshift. The fraction of major mergers for massive LTGs evolves faster (nMMLT ~ 4 ) than for ETGs (nMMET= 1.8). Conclusions: Our results show that massive ETGs have undergone 0.89 mergers (0.43 major and 0.46 minor) since z ~ 1, leading to a mass growth of ~30%. We find that μ ≥ 1/10 mergers can explain ~55% of the observed size evolution of these galaxies since z ~ 1. Another ~20% is due to the progenitor bias (younger galaxies are more extended) and we estimate that very minor mergers (μ < 1/10) could contribute with an extra ~20%. The remaining ~5% should come from other processes (e.g., adiabatic expansion or observational effects). This picture also reproduces the mass growth and the velocity dispersion evolution of these galaxies. We conclude from these results, and after exploring all the possible uncertainties in our picture, that merging is the main contributor to the size evolution of massive ETGs at z ≲ 1, accounting for ~50-75% of that evolution in the last 8 Gyr. Nearly half of the evolution due to mergers is related to minor (μ < 1/4) events. Based on observations made at the European Southern Observatory (ESO) Very Large Telescope (VLT) under Large Program 175.A-0839.