Aims. At intermediate redshift, galaxy groups and clusters are thought to impact galaxy properties such as their angular momentum. We investigate whether the environment has an impact on the galaxies’ stellar angular momentum and identify underlying driving physical mechanisms. Methods. We derived robust estimates of the stellar angular momentum using Hubble Space Telescope (HST) images combined with spatially resolved ionised gas kinematics from the Multi-Unit Spectroscopic Explorer (MUSE) for a sample of ∼200 galaxies in groups and in the field at z ∼ 0.7 drawn from the MAGIC survey. Using various environmental tracers, we study the position of the galaxies in the angular momentum–stellar mass (Fall) relation as a function of environment. Results. We measured a 0.12 dex (2σ significant) depletion of stellar angular momentum for low-mass galaxies (M_⋆ < 10¹⁰ M_⊙) located in groups with respect to the field. Massive galaxies located in dense environments have less angular momentum than expected from the low-mass Fall relation but, without a comparable field sample, we cannot infer whether this effect is mass or environmentally driven. Furthermore, these massive galaxies are found in the central parts of the structures and have low systemic velocities. The observed depletion of angular momentum at low stellar mass does not appear linked with the strength of the over-density around the galaxies but it is strongly correlated with (i) the systemic velocity of the galaxies normalised by the dispersion of their host group and (ii) their ionised gas velocity dispersion. Conclusions. Galaxies in groups appear depleted in angular momentum, especially at low stellar mass. Our results suggest that this depletion might be induced by physical mechanisms that scale with the systemic velocity of the galaxies (e.g., stripping or merging) and that such a mechanism might be responsible for enhancing the velocity dispersion of the gas as galaxies lose angular momentum.