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The direct discovery of gravitational waves (GWs) from the coalescence of compact binary components by the LIGO/Virgo/KAGRA Collaboration provides an unprecedented opportunity for exploring the underlying theory of gravity that drives the coalescence process in the strong and highly dynamical field regime of gravity. In this paper, we consider the observational effects of spatial covariant gravities on the propagation of GWs in the cosmological background and obtain the observational constraints on coupling coefficients in the action of spatial covariant gravities from GW observations. We first decompose the GWs into the left- and right-hand circular polarization modes and derive the effects of the spatial covariant gravities on the propagation equation of GWs. We find that these effects can be divided into three classes: 1) frequency-independent effects on GW speed and friction, 2) parity-violating amplitude and velocity birefringences, and 3) a Lorentz-violating damping rate and dispersion of GWs. With these effects, we calculate the corresponding modified waveform of GWs generated by the coalescence of compact binaries. By comparing these new effects with the publicly available posterior samples or results from various tests of gravities with LIGO/Virgo/KAGRA data in the literature, we derive the observational constraints on coupling coefficients of the spatial covariant gravities. These results represent the most comprehensive constraints on the spatial covariant gravities in the literature.