  We construct the explicit Euclidean scalar Green function associated with a
massless field in a higher dimensional global monopole spacetime, i.e., a
$(1+d)$-spacetime with $d\geq3$ which presents a solid angle deficit. Our
result is expressed in terms of a infinite sum of products of Legendre
functions with Gegenbauer polynomials. Although this Green function cannot be
expressed in a closed form, for the specific case where the solid angle deficit
is very small, it is possible to develop the sum and obtain the Green function
in a more workable expression. Having this expression it is possible to
calculate the vacuum expectation value of some relevant operators. As an
application of this formalism, we calculate the renormalized vacuum expectation
value of the square of the scalar field, $<\Phi^2(x)>_{Ren.}$, and the
energy-momentum tensor, $<T_{\mu\nu}(x)>_{Ren.}$, for the global monopole
spacetime with spatial dimensions $d=4$ and $d=5$.
