While reading the OP my very first thought on the subject was that America's Shield would win hands down. However, the comments that followed has left me wondering.
The shield had been weakened over time. Evidence suggests that it's damage threshold was well above Galactus' power output during the first Secret Wars, as it was damaged by the Beyonder but not fully destroyed. It is later dismantled by a strike from Thanos wielding the Infinity Gauntlet, which showed us it's full limit.
Later, the Serpent does the same thing, but he assaults it with magic. Because of the type of assault that he used against it, things become a bit murky. The shield was at that time weak against magic, however that doesn't mean that it was weak against non magical forces.
I do not believe that Galactus could destroy, or even damage the shield as it was written pre Secret Wars. Thor later rumples it, but again, King Thor was a magical being.
Since Adamantium should be vulnerable to the same forces that America's Shield was before the final upgrade, I'd vote that the shield would on average surpass the strength of Adamantium, but it wouldn't be by a lot.