Which group on the periodic table is naturally stable (unreactive) due to the fact that the atoms' valence shells are full?

Group 18: noble gases

This is a matter of memorizing the groups of the periodic table. The rightmost column is group 18 which is the noble gases: Helium, Neon, Argon, Krypton, Xenon, Radon. In general, the noble gases don't make chemical compounds. Although there have been some compounds made with the heavier noble gases (Krypton, Xenon, Radon).