Ran across a description of one, in the Mexican-American War, that seems much more ludicrously murderous than usual. One officer calling another one a liar, things get out of hand, etc. etc. but then:
Mumford announced that he would arrive with a musket, and that Mahan might bring any arms he thought proper. There were no seconds. Capts. Bankhead and Young were on the field, but only as mutual friends. 1st Lieutenant Thomas S. Garnett was also there to act in his professional capacity as physician. Both parties arrived on the field with muskets. In addition, Mumford wore a small dirk, and Mahan was equipped with a "five shooter", a pair of dueling pistols, and a large bowie knife. Attempts on the field to settle the matter amicably proved futile, and at about six o'clock the opponents took up their positions about sixty yards apart. Both then advanced about four paces, halted, aimed, then recovered. These motions were repeated, and at the third halt, when they were both about thirty five paces apart, the opponents fired simultaneously. Both fell and were carried back...mortally wounded.
Wallace, L. (1969). The First Regiment of Virginia Volunteers, 1846-1848. The Virginia Magazine of History and Biography, 77(1), 46-77. Retrieved June 3, 2021, from http://www.jstor.org/stable/4247453
In 1847 these would have been unrifled muskets. But still, shooting at each other with muskets at around 55 yards, then again at around 40, and again at 35 yards would have been awfully suicidal... .and presumably at some point after that the knives would come out? There was later a military tribunal, but would anyone normally feel compelled to stop something this stupid before it happened?
The encounter there is certainly an interesting one, although not quite the most murderous sounding I've read of. Although it isn't not a duel, it is certainly notable for being quite outside the realm of what many would call a 'proper' duel, lacking official Seconds, and sounding like it was more just an arrangement to fight to the death. It reminds me of the 1812 encounter between Hopkins and Floyd, where each had a pistol, a shotgun, and a knife, and could advance and shoot at will, although in their case there were Seconds, even if it was utterly absurd.
I digress though! Generally speaking, it was absolutely considered proper to stop a duel at certain points, but there were also points where it would be quite inappropriate. The encounter you detail here is most problematic in this regards as it lacked Seconds. Bankhead and Young are described as coming only in personal capacity, so there really isn't anyone to fill the role which could stop the encounter, as properly speaking, that was a key function of the Second. Many dueling codes, and advice for duelists, echo that of Lyde when he wrote that "nine duels out of ten, if not ninety-nine out of a hundred, originate in the want of experience in the seconds." His point being that the Seconds should always be working to bring about a conclusion to an affair of honor without resorting to the duel, and when a duel happens, it was because they failed in that regard.
Even if the duel did happen though, the Second ought to still be working to end the duel. Negotiations between the Seconds on the field could very well avert the duel before it happened, and it was only once the duelists had taken their marks that it would be inappropriate to try and end the duel, as at that point it would be considered cowardly. But between every exchange of fire - and this is where Mumford and Mahan's lacking of Seconds matters - the Seconds would be checking to see if honor was satisfied, and work to convince their man to agree to end the business without further exchange.
It was considered especially inappropriate to fire more than three shots in a duel, both because it was a sign of bloodthirstiness on the part of the duelists, but also because it reached the point of absurdity. In the event that three exchanged happened, it was almost always the case that both Seconds would insist on ending the matter regardless of their man's preference, and it was hardly unheard of for the Seconds to simply leave in protest if the duelists continued to insist. One of the most infamous cases of this was the case of Capt. Lee, who argued for an end of the business after three shots. Not only did his Principal disagree, but so too did the other duelist and his Second. Mr. Lee left the field in protest... and his man, Mr. Barrow, was shot dead on the next fire.
That of course is all very proper, and following 'the code' of what a proper duel should look like. The example you bring up obviously doesn't and as I stated at the beginning, some wouldn't even call it a duel. As with most things, duels existed more on a continuum of honor violence than a strict dichotomy, and this was especially true in the antebellum United States, where you get encounters like the 1827 "Sand Bar Duel" which ended up turning into a bloody melee involving the spectators, leaving several dead (numbers vary depending on the source, but at least two for certain) and a number more wounded. In that case, far from being a mediating factor, the Second were gleeful participants in the violence.
So anyways, as to your question, the key take away ought to be that while there were ways to stop a duel, not all duels were particularly apt to being stopped.