  Although recent Higgs data from ATLAS and CMS are compatible with a Standard
Model (SM) signal at $2\sigma$ level, both experiments see indications for an
excess in the diphoton decay channel, which points to new physics beyond the
SM. Given such a low Higgs mass $m_H \sim 125 {\rm GeV}$, another sign
indicating the existence of new physics beyond the SM is the vacuum stability
problem, i.e., the SM Higgs quartic coupling may run to negative values at a
scale below the Planck scale. In this paper, we study the vacuum stability and
enhanced Higgs diphoton decay rate in the Zee-Babu model, which was used to
generate tiny Majorana neutrino masses at two-loop level. We find that it is
rather difficult to find overlapping regions allowed by the vacuum stability
and diphoton enhancement constraints. As a consequence, it is almost inevitable
to introduce new ingredients into the model, in order to resolve these two
issues simultaneously.
