Could the legendary sword of Beleg Strongbow and Túrin Turambar, forged by Eöl of star iron, have done any damage to the One Ring?
I know the answer is most likely no but I just finished my re-reading of The Silmarillion and wanted to pose the question here because I'm not totally sure. It is made fairly clear that nothing but the fires of Orodruin can unmake the ring, but Gurthang is said to have properties of no other sword on Middle-earth.
Then Beleg chose Anglachel; and that was a sword of great worth, and it was no named because it was made of iron that fell from heaven as a blazing star; it would cleave all earth-delved iron. One other sword only in Middle-earth was like to it.
The Silmarillion, "Of Túrin Turambar", p. 247"
Linking Anglachel to Gurthang:
The sword Anglachel was forged anew for him by cunning smiths of Nargothrond, and though ever black its edges shone with pale fire; and he named it Gurthang, Iron of Death.
The Silmarillion, "Of Túrin Turambar", p. 258"
Just to add more to this question since the discussion thus far is that nothing indicates that it would just because it's a magical sword. I don't have my copy of The Silmarillion with me at the moment so I can't augment with more quotes unfortunately. But what makes the One so indestructible? Sauron. After instructing in the making of the 9, 7 and 3 he knew that he had to make the One great enough to be able to control and bind the others and thus he poured a great deal of his will and essence into the forging of the One.
From that I can conclude, perhaps mistakenly (which is where you fine people come in), that the One is only, perhaps, several magnitudes more powerful than Sauron at his peak.
It is discussed (I believe in the Council of Elrond) that no craft or power that they have can unmake the Ring and it needs to be cast back where it was forged, in the fires of the center of the Earth. I do not take that as evidence that only the fires of Orodruin can destroy it.
At the time of its making and up until Gandalf reappears as Gandalf the White there was no power in Middle-earth greater than Sauron - so it stands to reason that nothing short of Mt. Doom would be powerful enough to destroy it.
Now back to Gurthang. Gurthang is no ordinary sword. It is distinct from even the other notable swords of the Ancient World, of which we have master works of the Noldor forged in Valinor, perhaps even by Fëanor himself. (See above quote about "One other sword only in Middle-earth was like to it.") Apart from the quote about it being able to cleave all earth-delved iron, it is made of star iron. What is the origin of star iron? Was it devised by Varda when she created the stars? Or is it perhaps a remnant of Eru's own creation?
Also aside from the already named special qualities, it talks. It is the only item we hear about (apart from William's purse in The Hobbit) that physically speaks and throughout Túrin's story there are references to its emotion and will. This indicates a great inherent power in the sword.
There may be no answer and the answer may still be no, but I definitely think Gurthang is a fascinating part of the Ancient World and was curious to hear thoughts on its potential Ring slaying properties.