  We use resonant inelastic X-ray scattering (RIXS) at the Fe-L$_3$ edge to
study the spin excitations of uniaxial-strained and unstrained
FeSe$_{1-x}$S$_{x}$ ($0\leq x\leq0.21$) samples. The measurements on unstrained
samples reveal dispersive spin excitations in all doping levels, which show
only minor doping dependence in energy dispersion, lifetime, and intensity,
indicating that high-energy spin excitations are only marginally affected by
sulfur doping. RIXS measurements on uniaxial-strained samples reveal that the
high-energy spin-excitation anisotropy observed previously in FeSe is also
present in the doping range $0< x\leq0.21$ of FeSe$_{1-x}$S$_{x}$. The
spin-excitation anisotropy persists to a high temperature up to $T>200$ K in
$x=0.18$ and reaches a maximum around the nematic quantum critical doping
($x_c\approx0.17$). Since the spin-excitation anisotropy directly reflects the
existence of nematic spin correlations, our results indicate that high-energy
nematic spin correlations pervade the regime of nematicity in the phase diagram
and are enhanced by the nematic quantum criticality. These results emphasize
the essential role of spin fluctuations in driving electronic nematicity and
open the door for uniaxial strain tuning of spin excitations in quantum
materials hosting strong magnetoelastic coupling and electronic nematicity.
