No, this is based on a superficial reading of history. Toynbee was out to construct a grand unified theory of history. It's great fun but the finicky details just have to be swept under the rug for the theory to look impressive.
Let's look at it periodwise:
Early/Middle Republic: An agrarian society (see Tom Au's answer)
Late Republic: Rome was constantly expanding and conquering new lands, that is true. It was also woefully exploiting and mismanaging the new lands it had acquired and driving them into debt (check out the enter link description herepublicani). But was the expansion necessarily driven by plunder or the other way around? There is not real evidence for that, although it has long been a staple of Marxist historians.
Early Empire: No conquest, no plunder, no destitution of small-scale farmers. (Yes there is that famous quote "latifundies have destroyed Italy" but it's just that - good copy). Read Rostovzeff - the Empire was prosperous, the best time for people to live in - materially speaking - to live in till, say, the 16th century, at least. And yes, it was, to a large degree (but not completely!) a slave economy. So where did they get the slaves? They bred them. (See this paper for a dissection of the way this simple issue got muddled in the 19th century).
Late Empire: That's when the empire supposedly did turn into a too-heavy military machine relying for its maintenance on a huge bureaucracy that sucked the life out of the peasants. But as the previous period shows, this was not a structurally pre-determined situation.
And one more thing: the military machine built by Augustus and that was in place for all of the Early empire's duration (200 years) was actually a relatively small one; some historians even claim it was intentionally designed that way on order to disable future rulers from embarking on unbridled conquest, which Augustus thought could be very destabilizing. Others diagree, of course, about his intention, but there is little argument that the Principate's military establishment was in fact small and economical.
To sum again: Toynbee posited a theory that got only 2 of the 4 periods right and even there when you look at the data the fit seems rather contrived. But it's surely all great fun.