The existence of a kinematic morphology-density relation remains uncertain, and instead stellar mass appears the more dominant driver of galaxy kinematics. We investigate the dependence of the stellar spin parameter proxy $lambda_{R_e}$ on environment using a marked cross-correlation method with data from the SAMI Galaxy Survey. Our sample contains 710 galaxies with spatially resolved stellar velocity and velocity dispersion measurements. By utilising the highly complete spectroscopic data from the GAMA survey, we calculate marked cross-correlation functions for SAMI galaxies using a pair count estimator and marks based on stellar mass and $lambda_{R_e}$. We detect an anti-correlation of stellar kinematics with environment at the 3.2$sigma$ level, such that galaxies with low $lambda_{R_e}$ values are preferably located in denser galaxy environments. However, a significant correlation between stellar mass and environment is also found (correlation at 2.4$sigma$), as found in previous works. We compare these results to mock-observations from the cosmological EAGLE simulations, where we find a similar significant $lambda_{R_e}$ anti-correlation with environment, and a mass and environment correlation. We demonstrate that the environmental correlation of $lambda_{R_e}$ is not caused by the mass-environment relation. The significant relationship between $lambda_{R_e}$ and environment remains when we exclude slow rotators. The signals in SAMI and EAGLE are strongest on small scales (10-100 kpc) as expected from galaxy interactions and mergers. Our work demonstrates that the technique of marked correlation functions is an effective tool for detecting the relationship between $lambda_{R_e}$ and environment.
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