Context. The contribution of the merging process to the early phase of galaxy assembly at z > 1 and, in particular, to the build-up of the red sequence, still needs to be accurately assessed. Aims. We aim to measure the major merger rate of star-forming galaxies at 0.9 < z < 1.8, using close pairs identified from integral field spectroscopy (IFS). Methods. We use the velocity field maps obtained with SINFONI/VLT on the MASSIV sample, selected from the star-forming population in the VVDS. We identify physical pairs of galaxies from the measurement of the relative velocity and the projected separation (r p ) of the galaxies in the pair. Using the well constrained selection function of the MASSIV sample, we derive at a mean redshift up to z = 1.54 the gas-rich major merger fraction (luminosity ratio μ = L 2 /L 1 ≥ 1/4), and the gas-rich major merger rate using merger time scales from cosmological simulations. Results. We find a high gas-rich major merger fraction of 20.8 +15.2 −6.8 %, 20.1 +8.0 −5.1 %, and 22.0 +13.7 −7.3 % for close pairs with r p ≤ 20 h −1 kpc in redshift ranges z = [0.94, 1.06], [1.2, 1.5), and [1.5, 1.8), respectively. This translates into a gas-rich major merger rate of 0.116 +0.084 −0.038 Gyr −1 , 0.147 +0.058 −0.037 Gyr −1 , and 0.127 +0.079 −0.042 Gyr −1 at z = 1.03, 1.32, and 1.54, respectively. Combining our results with previous studies at z < 1, the gas-rich major merger rate evolves as (1 + z) n , with n = 3.95 ± 0.12, up to z = 1.5. From these results we infer that ∼35% of the star-forming galaxies with stellar masses M = 10 10 −10 10.5 M have undergone a major merger since z ∼ 1.5. We develop a simple model that shows that, assuming that all gas-rich major mergers lead to early-type galaxies, the combined effect of gas-rich and dry mergers is able to explain most of the evolution in the number density of massive early-type galaxies since z ∼ 1.5, with our measured gas-rich merger rate accounting for about two-thirds of this evolution. Conclusions. Merging of star-forming galaxies is frequent at around the peak in star formation activity. Our results show that gasrich mergers make an important contribution to the growth of massive galaxies since z ∼ 1.5, particularly on the build-up of the red sequence.