If you could kill off one of the scions...

[Cilia]

To correct myself, Midgardsormr never says he wants to stop the war; he simply refuses trying to do so because of the Ishgardians' blindness to their sin. It's not until you restore the Blessing of Light from his suppression that he believes there's a real chance for peace between man and dragon.

At the same time, he makes no indication that he's going to join Nidhogg, especially if you compare the Japanese and English dialogue. He simply states that he awoke in response to Nidhogg's call. While his English dialogue implies he's going to join the battle, in the Japanese script he never says anything of the sort and only speaks as if Nidhogg (the child in question) and his Horde are going to fight Ishgard.

Either way, he implicitly condemns Nidhogg's inability to let go of his hatred after he possesses Estinien to resurrect himself, so obviously Midgardsormr doesn't think that's kosher. Maybe it was, but he has decided to give Ishgard a chance even despite their grievous sins because he is rational enough to recognize the modern Ishgardians are not Thordan and thus do not deserve to suffer for his sins now that they know the truth and are trying to make up for it. Nidhogg... not so much. He hates Ishgard so much that he possessed a guy from beyond the grave and reconstituted his body though sheer hatred. Even Midgardsormr, Wyrmking and Magical Space Dragon, cannot condone this, which is why he warns Ishgard that Nidhogg will be back.

And, because Nidhogg does not recognize that the modern citizens of Ishgard are not responsible for Thordan's sin, or otherwise doesn't care, he will wage eternal war on them to see them suffer forever. No other reason. Compare Midgardsormr (who is willing to give them a chance) and Hraesvelgr (who doesn't care for the most part, though he does bring Ysayle to Azys Lla so maybe he had a change of heart).

Understandable? I... guess. Relatable? Hard. Justifiable? Not in the slightest.

Now, the problem is the time difference between man and dragon. Throughout the story, Hraesvelgr and Midgardsormr treat the incident with Thordan and his Knights Twelve as if it were yesterday to them. Fair enough, but Thordan and half the Knights were killed in recompense (already an unfair exchange if you follow the "eye for an eye" philosophy). Ishgard still stands, but the people who actually wronged the dragons are long dead. Midgardsormr says that the people of Ishgard have sinned, are blind to their sin, and want to sin yet again, but that's an extremely unfair judgment to make when even the Knight-Commander of the Temple Knights and the Azure Dragoon do not know the truth and are not in on the "Knights of the Round" plan. Remember that openly questioning the Church gets you branded a heretic, and that the only fate for heretics is death, and you'll understand why people don't question the Church's... embellished version of the story. (Hell, Estinien did have his doubts, he just never voiced them.)

So again, it comes back to the fact that Nidhogg hates the very concept of Ishgard and has no intention of actually winning the war that makes it necessary to put him down. To save the countless lives that would be consumed by his unyielding hatred, and for all the ignorant, innocent people he shamelessly killed... I did not feel good about it, but it had to be done and must be done once more. So be it.