Absolutely, though I would add a few important nuances to his argument. Be warned, I owe a great deal of my understanding of this period to Howard-Johnston's students and colleagues, so I guess I'm not the most unbiased person to answer this question. Howard-Johnston also has a volume focusing specifically on this conflict coming out in the near future, so I suspect there is room for a much fuller discussion on this topic in the next year or so.

To begin with, I think it is clear that the war of 603-628 was the culmination of the long-term rivalry between Rome and Persia, a conflict that saw Ctesiphon reach the height of its power, essentially ruling the wealthiest parts of the eastern Mediterranean world by 619, whilst Constantinople fell to its lowest ebb, with the emperor literally offering to become a vassal of the mighty Khusro II in 615. The rivalry between these two great empires, one that had characterised much of late antiquity, now reached its bitter conclusion, as both powers, driven by the ambitions of their talented rulers, fought a war on a far grander scale than ever before. No matter what we think of it as, it was a war on a truly grand scale, one that drew in powers ranging from the Avars in the Balkans to the Turks in the Eurasian steppes.

But was it the last? Increasingly, historians have understood the Arab conquests as a very late antique phenomenon and as such I don't think it can be viewed as the beginning of a new era. Instead, the conquests' successes, and indeed the cultural and ideological themes it embodied, was largely due to the turmoil of the early seventh century. Apocalyptic sentiments in the Near East reached an all time high, holy war was preached from the pulpits, and disaffected Arab chiefs, their power already evident in the revolts of the sixth century, were now ranging far and wide to grab for themselves a slice of the imperial pie even as early as the 610s. In my opinion, I think it is an ultimately fruitless to point to a date and say that's when things changed, so I prefer instead to identify a broad period of time when the patterns visible in late antiquity perhaps shifted in a more medieval direction. I personally would suggest c.650-700 for when that happened, though it was a subtle change and not understood as the dawn of the new world until sometime in the eighth century. It gets more complicated when we bring in western Europe, which is generally seen as having entered the medieval period well before then. Things like personal beliefs and forms of economic exchange simply never changed as dramatically as historians would like them to, nor is it possible to satisfactorily distinguish between different historical epochs. Periodisation is a difficult issue to unravel and there are many different views out there, so I also see no reason to be particularly loyal to a specific definition.

As for how this ties in into Howard-Johnston's formulation, I think it depends on what you see as more important. For me, although I don't think that we can divide the seventh century neatly into its late antique and early medieval phases, I do recognise that the political order, as represented by Rome and Persia as the 'two eyes of the world', had changed. Both had suffered grievous wounds and so both focused instead on recuperating their health in the war's aftermath. For Heraclius, it meant the implementation of his grand strategy to reunify the many different branches of Christianity, whilst in Persia different dynasts competed with each other to seize the throne. Conflict between the two empires was, I think, on hold for the forseeable future.

Furthermore, the genie had now been let out of the bottle and the consequences of this great conflict, its legacy of holy war, warrior emperors, and religious polemics, could be not be forgotten so easily. Viewed from this perspective, future conflicts simply cannot return to the border wars of the sixth century, a time when neither empires were able to win convincing victories over the other, leaving both to instead scrap over remote fortresses and to battle via their frontier clients. In political terms, I would argue that wars between these two powers could never be the same again, even if the Arab conquests did not destabilise the system further. In this sense, the war of 603-628 really was 'the last great war of antiquity', though as I've noted in other ways it was merely the first stage of a 'world crisis' that would engulf the Near Eastern world for the next century.

Interestingly enough, a contemporary Armenian historian, pseudo-Sebeos, thought about his troubled world in a similar way. From the prologue of his History, which covered events up to c.660:

Now I shall recount in narrative fashion the tale of the destructive events which befell the world: the wrath evoked from on high and the anger flaming up below; the torrents of fire and blood, and the raids of brigands; the death-bringing attacks, the cry of demons and the roar of dragons, the races of Chaldaeans and of men descended from giants, of brave armed cavalry, from east to west, from north to south. [I shall describe] those of the south, aroused with great passion, attacking each other, and the fulfilment of the command of the Lord’s anger against the whole world. Like the whirlwind they arose and burst out to destroy everything within, to raze mountains and hills, to rend the plains, to crush in pieces the stones and rocks beneath the heels of their horses and trampling hooves.

So I shall expound the story of the destructive and ruinous Khosrov, cursed by God.

Though pseudo-Sebeos was a historian well aware of the problems closer to his own time, he still nonetheless pointed to Khusro II's ambition as the primary cause of both his war with Rome and later the Arab conquests. This signals not just his awareness of the uniqueness of Khusro's war, but also how closely related the events after war were to the chaos of the previous decades. Admittedly pseudo-Sebeos approached it from a rather apocalyptic and anti-Persian/Arab perspective, rather than from a purely historical viewpoint, but it is still worth thinking about his words and how contemporaries viewed events, in addition to how modern historians have tried to narrate this fascinating century.

Certainly if one considers the Muslim conquest of the Sassanid empire as the "end of antiquity" and the beginning of the early medieval era. It is the last great war because the next one defined the end of the era.

Then the question becomes is the Muslim conquest of the Sassanid Empire the best delimiting event between late antiquity and early medieval. There's a lot of good points stating that it is, since the fall of the Sassanid empire meant the Byzantine Empire was then the only surviving great power from classical antiquity. All of the other powers of the era are relatively new. The Muslim conquest of North Africa and Levant also split off those regions from what the classical Roman world. A parallel can be drawn to how the fall of the Soviet Union marked the end of the cold war.

Related question: what's the most widely used name for the Neo-Persian Empire? Sassanid? Sassanian? One s or two? Wikipedia uses Sasanian but I'd previously only encountered Sassanid.