This is an old idea that goes back quite far, and while there's more than a little truth to it it obscures rather a lot. I've written on a similar version of the question here.

The problem with the Roman empire, from a political science or world history point of view, is that almost the entirety of Roman imperial expansion took place while the state was still a Republic. The Roman Republic was of course not what we mean by "republic," in that despite the recent (and I believe fundamentally correct) trend to show that the Republic was not as oligarchic as was generally thought sixty years ago, it was certainly not a democracy. Nevertheless, it was not an autocracy of the sort that we might normally expect from other "great empires," ancient and modern. This problem extends to its economy as well. Like the Athenian Empire, which maintained its citizens' democratic rights through the collection of tribute from its subjects, the rights and privileges of Roman citizens in the Republic were defined in opposition to and maintained by provincial conquest. Why this is so is a problem all scholars of Roman imperialism have had to deal with. Until recently the standard study was William Harris's, which postulated that the Romans were simply unusually aggressive, and that if anything the Republican government encouraged such a culture. In 2006 Eckstein blew the doors off Roman imperialism by pointing out that Roman culture cannot be shown to have been more bellicose than those surrounding it. Eckstein combined Harris' observation that the Republic was essentially constantly at war with previous scholars' points regarding the general lack of an "imperial program" in the Republic to suggest that the Republic found itself a player in a state of anarchy spanning the Mediterranean, and that the conquest of the Mediterranean was a natural result of the lack of state centrality in the region.

Eckstein's argument has raised a lot of questions and arguments, but it's shown that Harris' model is hard to support. With it goes also the economic model of a state maintained by conquest. However important conquest was in providing Roman society with slaves and booty, these cannot be considered sustainable activities, even despite the near-constant state of warfare during the Republic. For one thing, the armies of the Principate engaged in few genuine wars of conquest, yet the economy under the emperors was more robust than ever before. Slaves and plunder taken in raids across the frontier did not adequately serve to make up for the massive gains of taking entire provinces during the Republic. Moreover, you'd be hard-pressed to show that plunder and slaves taken by conquest drove the state generally. Polybius says that plunder was divided among the troops after a battle, and while the treasury took part of the spoils the majority of captured wealth would have been distributed in penny packets to individual soldiers. This might have enriched them when returning home--perhaps especially if Rosenstein's model of extended families on small farming plots relieving population pressure by sending their sons to war is correct--but this cannot be considered anything but a small shot in the arm to the economy.

The other part of the question is tribute. Active conquest was not necessary for the supply of tribute, and tribute made up the vast majority of state revenue during the Republic and much of the Principate. There can be no question that this was made possible by conquest. However, the changes to citizenship law and taxation in the early third century did away with this, and the economy forged on, weakened by the political instability of the third century but probably not by significant problems with the system. There is no evidence to suggest that the Romans ever outstripped the supply of provincial tribute, necessitating more tribute (which would in turn require more armies and more costs). So the need to acquire more tribute doesn't seem to be relevant.

The crucial thing is that the Roman economy was an agrarian society. It was also uniquely aided by the Mediterranean, which served to connect distant points with relative ease of travel. The establishment of a single ruling structure over the Mediterranean allowed for much greater and safer movement of trade, which can be pretty easily shown. In this respect, then, the conquest of the provinces was absolutely necessary to the strength of the economy. But as my linked comment shows, there was much more to the Roman economy than taking in slaves and plunder from conquest. The reverse appears true, namely that the major economic activity of the Roman world was trade which was bolstered from time to time at the lower and treasury level by conquest, not the other way around. Of course, this was made possible by imperial expansion, but that's not the same thing as saying that the economy was expansion.