We don't know that much about dittany.
Dittany is only used in the series a few times. It is mostly used in Deathly Hallows on an assortment of injuries, most notably Ron's splinching. Hermione also uses on Harry's snake bite after the Godric's Hollow fiasco:
The snake bit you too but I've cleaned the wound and put some dittany on it...."
And they use it on their burns after the Gringotts break-in:
Both had angry red burns all over their faces and arms, and their clothing was singed away in places. They were wincing as they dabbed essence of dittany onto their many injuries.
Dittany was also recommended for Malfoy's wounds inflicted by Harry's Sectumsempra in Half-Blood Prince:
"You need the hospital wing. There may be a certain amount of scarring, but if you take dittany immediately we might avoid even that.... Come...."
This doesn't tell us very much about the parameters of dittany's uses. In Ron's case it was apparently able to restore nonexistent flesh, while in the other cases it seems to have only done minor healing. Notably, in Malfoy's case the wounds were inflicted by Dark magic and that doesn't seem to have been an obstacle for dittany. On the other hand, in that case dittany was mainly being used to prevent scarring rather than as the actual healing agent.
Thus, it is certainly possible that dittany could have worked for Moody's nose, but I don't think we can assert it definitively without having more information on the nature of dittany and what caused the injury. We know from Goblet of Fire that the injury was inflicted when Rosier was killed:
"Rosier is dead," said Crouch. "He was caught shortly after you were too. He preferred to fight rather than come quietly and was killed in the struggle."
"Took a bit of me with him, though," whispered Moody to Harry's right. Harry looked around at him once more, and saw him indicating the large chunk out of his nose to Dumbledore.
But we don't know what precisely removed the chunk of his nose.
It also seems from some of the aforecited instances of dittany use that it may be important for the dittany to be applied quite soon after the injury is incurred. In the case of the splinching and in the case of the snake bite the dittany was applied immediately. In the case of the burns it was applied within a couple of hours. And in Malfoy's case Snape specifically mentioned that it should be applied immediately.
Thus, it is also possible that dittany could theoretically have healed Moody's injury, but only if applied soon enough. Depending on the circumstance of the altercation with Rosier (which we don't know much about), it is possible that Moody simply didn't have access to dittany until it was too late.
He may have deliberately kept his injuries.
We can also suggest an entirely different possibility. Perhaps dittany would have worked but Moody simply didn't want to heal his injuries. This might seem a bit strange at first, but consider Dumbledore's broken nose. This injury appears to have been a pretty minor injury inflicted by mundane means, as described in Deathly Halllows:
"His heart wasn't the only thing. Didn't Aberforth break Albus's nose halfway through the service?"
The way Bathilda told it, Aberforth shouted that it was all Abus's fault that Ariana was dead and then punched him in the face.
So Dumbledore's nose was apparently broken by a simple punch to the face. This should have been easily reparable, as Tonks was able to heal Harry's broken nose in Half-Blood Prince with a simple spell:
And he stamped, hard, on Harry's face. Harry felt his nose break; blood spurted everywhere.
"Episkey," said Tonks.
Harry's nose felt very hot, and then very cold. He raised a hand and felt it gingerly. It seemed to be mended.
So it would seem from this that Dumbledore deliberately let his nose remain broken for the rest of his life, presumably as some sort of reminder/badge of shame of what had happened.
In a somewhat similar vein we could suggest that Moody deliberately did not heal his injuries so that they would serve as a reminder/badge of honor of all he did to fight Dark magic. Each additional scar and injury would be another source of pride. The scars and injuries would also give him a certain "battle-worn look" that would announce to opponents that he is not someone to be messed with.