I've read most of his book, and it's crap.
The west was not superior at warfare, in any time. The Greek poli were able to defeat the Persians, why? Because, for all of the citizen farmer rhetoric, they were called up so often, they were essentially professional soldiers drawn up against the unmotivated levies of Persia. When they actually ran up against forces which were actually decent soldiers, like Central Asia, they took years to subdue it, or in India, which they never attacked at all. You may say for all you wish that he turned back because his soldiers wouldn't follow him any further; that argument is BS. Indian armies were powerful enough to make him think twice--a relatively minor king of Porus was defeated because he had the good luck to come during the monsoon. If he had come any further, the Nanda armies would have destroyed his army piecemeal.
Let's go onto the next most cited example. Rome. Rome was... hmm... how shall we say it? Remarkable. But not because of its culture, or because of its cultural ethics. Rome was a born and bred killing machine, who paid the millions of people that they garnered from the rich levies of Italy with the gold that they pillaged from the land. Nothing that was not militarily significant or had practical purpose was developed by the Romans; they had roads to have their armies travel long distances, aqueducts to water their cities, walls to keep out enemies. What is not pointed out is that the Romans have exactly what Hanson detests--essentially a theocratic despotism, the emperor being a god, with the empire built on thousands of slaves and the pre-medieval equivalent of a fuedal society. There was no proto-capitalism, there was none of that BS he brings up all the time; Rome was in and of itself a remarkable state that was built solely to conquer other states.
From thereon we see a medieval Europe fall into the Dark Ages, personally my favorite period of Western History, but also the point where it is utterly impossible to argue that the West was any good at all in warfare. They managed to repel the Arabs at Poitiers? Sure, but that ignores the fact that the Arabs were at Poitiers at all. They arrived tired after a long march... A long march, folks. The West wasn't doing any triumphant conquering, they were repelling the mere vanguard of Arab invasion. Yes, this stopped the Arab advance from spreading any further. Anyone want to tell me when the Muslims were actually evicted from Spain? Oh, right, 1492. How about from Europe in general? Oh, right, they're still there. Fact is, the West was invaded, and it took them centuries to repel the invasion.
Crusades! you might boldly cry out. Right, Crusades. Any volunteers as to what happened to the Crusader states? They died, to put it bluntly. They invaded in a time of internal Muslim squabbles, survived because the Mongols essentially flanked the Muslims on a massive scale, and the Muslims were fighting the Byzantines as well. And yet, it took a mere two hundred years for them to drive out the West (compare this to 700 years for the expulsion of the Moors and still pending for the Turkish retreat).
By 1400, we see the emerging synthesis of Western armies... which still don't do much at all. Their cannonry is inferior to that of the Turks, the Indians, and even the Chinese. They adopted it from the Chinese, in fact, who invented it two centuries earlier. Western armies were still not even on par with the Turkish slave armies Hanson derides all the time.
The Reformation is probably the catalyst for Western fast expansion, in that it divided Europe into two competing parts, which developed technology much faster. And yet, in and of themselves, they still didn't have much of a military edge on the rest of the world. between 1400-1700, the main Western asset was in that they could fund a war with American riches, and that this could buy a highly disciplined army. After this time is when we finally see the Western armies clearly superior.
Overall, then, the West was not superior for most of its history. It became superior due to the fact that its geography discouraged invasion, but still permitted information exchanges, so they recieved the technology of elsewhere, and could develop it on their own. It also prospered due to its fortunate geographic location: the Americas, a vast, virgin land, easily exploitable, that no other civilized power was nearby. The riches of the Americas funded the Hapsburg domination of much of Europe, prompting the alliance against it. American riches funded the beginning of the British Empire, allowing them to take over India with superior discipline, so India could take the new place of the crown jewel of the British Empire.
It was geography, and a lot of luck in the great commanders, inventors, and leaders they had. They dodged the bullet so many times, it's not even funny. The West was not superior, merely lucky.