How about something Image3D-based?
grid = First@
ParametricPlot3D[{Cos[u] Cos[v], Sin[u] Cos[v], Sin[v]}, {u, 0,
2 Pi}, {v, -Pi/2, Pi/2}, PlotStyle -> None];
Graphics3D[
{grid,
{Red, Thick, Line[1.2 {{-1, 0, 0}, {1, 0, 0}}]},
{Blue, Thick, Line[1.2 {{0, -1, 0}, {0, 1, 0}}]}, {Green, Thick,
Line[1.2 {{0, 0, -1}, {0, 0, 1}}]},
Raster3D[Array[
UnitStep[1 - Norm[{##}]] &, {100, 100,
100}, {{-1., 1.}, {-1., 1.}, {-1., 1.}}], {{-1, -1, -1}, {1, 1,
1}}, ColorFunction -> (GrayLevel[0.8, .03 #] &)]
},
Boxed -> False]
