The previous answers offer a good explanation of the book canon, but since your question is clearly inspired by the Great Eye of the movies, I'll make the distinction more explicit:
The colossal, fiery "Great Eye" is a creation of the movies...
In the book, it's mostly a metaphor and a logo for Mordor, while Sauron has a normal physical body. He never appears directly in any scene of the book, he's only mentioned and described, which is why the movies were able to re-imagine him as a more concrete "monster", rather than the harder-to-portray psychic presence he takes in the books, without changing much.
So in the book, there's no question: he just would have put on the Ring, as anyone does.
...but he probably had a physical body there, too.
The final cuts (and extended cuts) of the movies never actually comment on whether there's a physical Sauron in addition to the Great Eye (sort of like in the Wizard of Oz). But most interestingly, there was a deleted scene in the films that involved Aragorn fighting Sauron personally outside the Black Gate during the climax of Return of the King:
That scene ended up getting replaced by the fight with the Battle Troll, because the filmmakers decided that "those 10-foot monster fights" tend to come across as silly, and it drew attention away from Frodo's final showdown, which was actually the more important event in that moment.
But since Sauron had a normal body in the books and was supposed to appear with a normal body onscreen, I think we can safely assume that he had a normal body in the final cut of the film as well, regardless of whether we ever actually see it.
Therefore, both in the books and in the movies, Sauron would have simply worn the ring on his normal, albeit unusually large and claw-y and altogether evil, finger.
EDIT: For those doubting it's Sauron
There has been some debate in the comments about whether the screenshot I posted is of Sauron, or the Mouth of Sauron. There IS a scene in the Extended Editions with the Mouth of Sauron, but I am talking about a specific thing they discusses in the Extended Edition documentaries where Aragorn fought Sauron himself.
Originally, Sauron would have appeared as Anatar, the beautiful disguise Sauron wore when he tricked the Elves into making the Rings in the first place. Aragorn would have let his guard down (this is the moment in the final cut when the Great Eye sort of beams-in on Aragorn and he slowly lowers his sword), before Sauron transformed into his evil form and attacked:
I was able to find this rough-cut footage of the scene, along with a few more stills: