It wouldn't - but it would reduce Harry's risk of losing a finger to a Finite Incantatem in combat. And Harry does take pains to physically separate the ring from the stone before Dumbledore inspects the stone:

Slowly, Harry did so, removing the gem and setting the ring off to the other side of the desk.

Unless that's just an intended second red herring, to make Dumbledore feel even worse about mistrusting Harry a second time. But just like Harry, Dumbledore, McGonagall, and Snape err in completely distrusting the wards when they say the Defense Professor killed Hermione, we can't treat possibly falsified evidence as anti-evidence, this has to shift our belief at least a little bit toward the Ringmione hypothesis.

The ring is also a significantly less suspect place to hide Hermione than the stone, because transfiguring something into a ring that you wear goes directly contrary to McGonagall's safety advice. Partial transfiguration provides a way around the safety issue, and Dumbledore and McGonagall are unused to thinking about things that can be done with partial transfiguration, so the idea is less likely to occur to them.

I'm about 50% confident that Hermione is in the ring.

(By the way, the wards' identification of Hermione's killer as the Defense Professor is strong evidence for the theory that the Defense Professor was carrying some nasties standing in his pocket when Dumbledore drew a circle around Quirrel to mark the Defense Professor to Hogwarts's wards.)