  In this work, the approximate bound state solutions of the fractional
Schr\"odinger equation under a spin-spin-dependent Cornell potential are
obtained via the convectional Nikiforov-Uvarov approach. The energy spectra are
applied to obtain the mass spectra of the heavy mesons such as bottomonium,
charmonium and bottom-charm. The masses for the singlet and triplet spin
numbers increase as the quantum numbers increase. The fractional Schr\"odinger
equation improves the mass spectra compared to the masses obtained in the
existing literature. The bottomonium masses agree with the experimental data of
the Particle Data Group where percentage errors for fractional parameters of
\b{eta}=1,{\alpha}=0.97 and \b{eta}=1,{\alpha}=0.50 were found to be 0.67% and
0.49% respectively. The respective percentage errors of 1.97% and 1.62% for
fractional parameters of \b{eta}=1,{\alpha}=0.97 and \b{eta}=1,{\alpha}=0.50
were obtained for charmonium meson. The results indicate that the potential
curves coupled with the fractional parameters account for the short-range gluon
exchange between the quark-antiquark interactions and the linear confinement
phenomena which is associated with the quantum chromo-dynamic and
phenomenological potential models in particle and high-energy physics
