  A brane-world $SU(5)$ GUT model with global non-Abelian vortices is
constructed in six-dimensional spacetime. We find a solution with a vortex
associated to $SU(3)$ separated from another vortex associated to $SU(2)$. This
$3-2$ split configuration achieves a geometric Higgs mechanism for $SU(5)\to
SU(3)\times SU(2)\times U(1)$ symmetry breaking. A simple deformation potential
induces a domain wall between non-Abelian vortices, leading to a linear
confining potential. The confinement stabilizes the vortex separation moduli,
and assures the vorticity of $SU(3)$ group and of $SU(2)$ group to be
identical. This dictates the equality of the numbers of fermion zero modes in
the fundamental representation of $SU(3)$ (quarks) and of $SU(2)$ (leptons),
leading to quark-lepton generations. The standard model massless gauge fields
are localized on the non-Abelian vortices thanks to a field-dependent gauge
kinetic function. We perform fluctuation analysis with an appropriate gauge
fixing and obtain a four-dimensional effective Lagrangian of unbroken and
broken gauge fields at quadratic order. We find that $SU(3) \times SU(2) \times
U(1)$ gauge fields are localized on the vortices and exactly massless.
Complications in analyzing the spectra of gauge fields with the nontrivial
gauge kinetic function are neatly worked out by a vector-analysis like method.
