  A nodeless superconducting (SC) gap was reported in a recent scanning
tunneling spectroscopy experiment of a copper-oxide monolayer grown on the
Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ (Bi2212) substrate [Y. Zhong {\it et al.},
Sci. Bull. {\bf 61}, 1239 (2016)], which is in stark contrast to the nodal
d-wave pairing gap in the bulk cuprates. Motivated by this experiment, we first
show with first-principles calculations that the tetragonal CuO (T-CuO)
monolayer on the Bi2212 substrate is more stable than the commonly postulated
CuO$_{2}$ structure. The T-CuO monolayer is composed of two CuO$_2$ layers
sharing the same O atoms. The band structure is obtained by first-principles
calculations, and its strong electron correlation is treated with the
renormalized mean-field theory. We argue that one CuO$_2$ sublattice is hole
doped while the other sublattice remains half filled and may have
antiferromagnetic (AF) order. The doped Cu sublattice can show d-wave SC;
however, its proximity to the AF Cu sublattice induces a spin-dependent
hopping, which splits the Fermi surface and may lead to a full SC gap.
Therefore, the nodeless SC gap observed in the experiment could be accounted
for by the d-wave SC proximity to an AF order, thus it is extrinsic rather than
intrinsic to the CuO$_2$ layers.
