  In the previous papers, we studied the 't Hooft-Polyakov (TP) monopole
configurations in the U(2) gauge theory on the fuzzy 2-sphere,and showed that
they have nonzero topological charge in the formalism based on the
Ginsparg-Wilson (GW) relation. In this paper, we will show an index theorem in
the TP monopole background, which is defined in the projected space, and
provide a meaning of the projection operator. We also extend the index theorem
to general configurations which do not satisfy the equation of motion, and show
that the configuration space can be classified into the topological sectors. We
further calculate the spectrum of the GW Dirac operator in the TP monopole
backgrounds, and consider the index theorem in these cases.
