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In this letter, we investigate the impact of environment on integrated and spatially-resolved stellar kinematics of a sample of massive, quiescent galaxies at intermediate redshift ($0.6<z<1.0$). For this analysis, we combine photometric and spectroscopic parameters from the UltraVISTA and Large Early Galaxy Astrophysics Census (LEGA-C) surveys in the COSMOS field and environmental measurements. We analyze the trends with overdensity (1+$δ$) on the rotational support of quiescent galaxies and find no universal trends at either fixed mass or fixed stellar velocity dispersion. This is consistent with previous studies of the local Universe; rotational support of massive galaxies depends primarily on stellar mass. We highlight two populations of massive galaxies ($\log M_\star/M_\odot\geq11$) that deviate from the average mass relation. First, the most massive galaxies in the most under-dense regions ($(1+δ)\leq1$) exhibit elevated rotational support. Similarly, at the highest masses ($\log M_\star/M_\odot\geq11.25$) the range in rotational support is significant in all but the densest regions. This corresponds to an increasing slow-rotator fraction such that only galaxies in the densest environments ($(1+δ)\geq3.5$) are primarily (90$\pm$10\%) slow-rotators.This effect is not seen at fixed velocity dispersion, suggesting minor merging as the driving mechanism: only in the densest regions have the most massive galaxies experienced significant minor merging, building stellar mass and diminishing rotation without significantly affecting the central stellar velocity dispersion. In the local Universe, most massive galaxies are slow-rotators, regardless of environment, suggesting minor merging occurs at later cosmic times $(z\lesssim0.6)$ in all but the most dense environments.