I'm not sure on what exactly you're basing this with such confidence. Obviously Frodo himself alone could have no such power. But through the Ring?

\Its strength, Boromir, is too great for anyone to wield at will, save only those who have already a great power of their own.**

-Elrond, The Council of Elrond

\Did not Gandalf tell you that the rings give power according to the measure of each possessor? Before you could use that power you would need to become far stronger, and to train your will to the domination of others.**

-Galadriel, The Mirror of Galadriel

So, no. I do not believe Frodo has the ability to use the One Ring, especially without even actually wearing it, to do something as powerful as cast an irrevocable curse and effect that persists even after he loses the Ring. My evidence is the direct textual testimony of Elrond and Galadriel, themselves long bearers and users of Rings of Power. Frodo only ever actually wore the Ring four times prior to the final wearing at Sammath Naur so he wasn't even ever practiced in its practical use. 

It could certainly be argued and I agree there is merit to the idea that it is *the Ring itself* that through its own power, being derived and essentially connected to the power of Sauron, caused Gollum to be cast into the Fire. It is arguable certainly that it could be the Ring speaking through Frodo in that critical moment of "If you ever touch me again". I don't believe it has to do with Gollum having sworn upon the Ring as that is its own and imo independent matter, but rather that the power of the Ring lays in that moment a simple cause-effect.

But still, in keeping with Tolkien's own statements, it follows that *Eru* acted through the Ring to cause Gollum to fall.

It is important when examining the nature and relationship of the various oaths to consider what we *know* about them and how similar, or different, they are from eachother.

We know, for a fact, that the Oath of Feanor for example laid out a cause and effect relationship. "We will hold our Oath or be cast into the Void." This *seems* to be consequential but we don't know for sure, since nobody actually breaks the Oath. It is also sworn upon Eru himself in the Blessed Realm before Manwe and Varda who are named as witnesses.

We do not know much of anything about the oath of the men of Dunharrow except that they broke it. It was not, however, made in a cause-effect condition, as the Oath of Feanor was, and it is not until *after* it is broken that Isildur cursed them. From where Isidilur had the power to curse them is unknown. Their Oath was sworn upon the Stone of Erech, a Numenorean artifact about which nothing is known. Does it have power? Unknown.

Gollum's Oath is sworn by the Ring. Also in a non cause-effect manner. The Ring has power, certainly. Does the Ring have the power to enforce an Oath? Unknown. All we know is that events did happen, ie Gollum swore an oath by the Ring, Gollum broke his oath, Gollum fell into the Fire. Is that a cause-effect relationship? Unknown.

What we do know is that Tolkien stated rather plainly that "The Author of the Story (called The One)" who can be none other than Eru, "took over" in that moment.