  We study a prediction on neutrino observables in a non-supersymmetric
renormalizable $SO(10)$ GUT model that contains a ${\bf 10}$ complex scalar
field and a ${\bf 126}$ scalar field whose Yukawa couplings with ${\bf 16}$
matter fields provide the quark and charged lepton Yukawa couplings, neutrino
Dirac Yukawa coupling and Majorana mass for the singlet neutrinos. The $SO(10)$
breaking is achieved in two steps by a ${\cal O}(10^{15})$ GeV VEV of a ${\bf
54}$ real scalar field and a ${\cal O}(10^{14})$ GeV VEV of the ${\bf 126}$
field. First, we analyze the gauge coupling unification conditions and
determine the VEV of the ${\bf 126}$ field. Next, we constrain the Yukawa
couplings of the ${\bf 10}$ and ${\bf 126}$ fields at the scale of the ${\bf
126}$ field's VEV from experimental data on quark and charged lepton masses and
quark flavor mixings. Then we express the active neutrino mass with the above
Yukawa couplings and the ${\bf 126}$ field's VEV based on the Type-1 seesaw
mechanism, and fit neutrino oscillation data, thereby deriving a prediction on
poorly or not measured neutrino observables. What distinguishes our work from
previous studies is that we do not assign Peccei-Quinn charges on visible
sector fields so that the ${\bf 10}$ scalar field and its complex conjugate
both have Yukawa couplings with ${\bf 16}$ matter fields. From the fitting of
neutrino oscillation data, we find that not only the normal neutrino mass
hierarchy, but also the inverted hierarchy can be realized. We also reveal that
in the normal hierarchy case, the Dirac CP phase of the neutrino mixing matrix
$\delta_{CP}$ is likely in the ranges of $-2.4<\delta_{\rm CP}<-1.2$ and
$1.2<\delta_{\rm CP}<2.4$, and not in the region with $\delta_{\rm CP}\sim\pi$,
and that in the normal hierarchy case, $\theta_{23}$ is likely in the upper
octant and in the range of $0.50\lesssim\sin^2\theta_{23}\lesssim0.55$.
