There is an enormous and by now quite old debate about this that has resurfaced recently with the emergence of the "history of capitalism" literature . The old debate centred largely around Fogel and Engerman's "Time on the Cross", whereas the new debate was spurred by Sven Beckert's "Empire of Cotton" and Edward Baptist's "Half Has Never Been Told," among others.

The claim made in the latter literature is, broadly, that slavery was crucial not only for building up the American economy, but also the broader rise of global capitalism led by European industrialization. This interpretation is roundly rejected by most economic historians. Slavery was, in the estimation of most of the economic historians of American slavery (Gavin Wright, Alan Olmstead and Paul Rhode, to name the most prominent) a tremendously profitable institution for those who owned slaves; "capitalist" in the sense of managed, operated for profit, and innovative; and a major engine of the southern agricultural economy. But in the long run, slavery was an impediment to industrialisation that kept resources locked in relatively unproductive sectors with few long-term prospects, and prevented the broad spread of free labour and education that underpinned late 19th c. economic development. It was Northeastern industry rather than Southern cash crop agriculture that "built" America in terms of its "meteoric climb."

The "naive" case, that slavery was simply so widespread and enormously profitable that it spurred economic super-growth, is not plausible on the grounds of magnitude. Cotton was a sustantial sector, but certainly not large enough to have driven the economy overall, at only about 5% of GDP. There was simply not enough wealth generated through this process to fund industrialisation, and there is essentially no evidence that this was the case. Slave labour was not overwhelmingly cheaper than free labour, since slaves needed to be fed, clothed and housed, and certainly not cheap compared with labour elsewhere in the world. Even if we do consider the possibility of wealthy Southerners driving growth, in general, the slaveowners of the South were debtors rather than creditors, and it is not clear how they could have funded the development of industry in the North if capital was largely flowing the other way.

The relative position of the South in the US economy was deteriorating rather than improving from independence onward. At the time of the revolutionary war, the South was substantially richer than the North, but their lead was eroded during the early 19th century. If slavery was (by the time of industrialization) really that overwhelmingly productive compared with northern free labour, the South probably would have stood a better chance in the US Civil War. It wasn't, and they didn't, as Rhett Butler famously points out in Gone With the Wind.

A more sophisticated case is that slavery was important as a provider of cheap inputs (cotton, mostly) into the industrial process. This, however, fails to consider the alternatives in a counterfactual world. Free labour could (and after abolition, did) produce cotton in the US South at the same or even higher levels of productivity as under slavery. Cotton could also be produced elsewhere in the world than the US, including both places with slavery (Northern Brazil) and places with different labour practices (Egypt, India, Turkey). British producers had a surprisingly easy time adapting to the supply shock of the US civil war, and most of the adaptation required was to adapt machinery to different staple cotton rather than a massive price shock or a persistent shortage of inputs. Slave owners used slaves to produce cotton in the mid-19th century because that was a profitable crop to grow on their land, because of the enormous demand for cotton from (mostly) Britain - not the other way around, that slavery provided an otherwise impossible quantity of cotton at cheap prices.

So, in sum, the mainstream view from economic history is: Slavery was horrible, oppressive, and very profitable for a handful of people on top of the system of racial oppression in the south. It was more important for the late 18th and early 19th centuries than by the immediate antebellum period. Some elements of modern "capitalist" management, accounting and business practice were being developed on slave plantations - though also in other places and sectors. But slavery was, if anything, an active impediment to the real long run sources of growth in the United States, which required free(-ish) labour markets, educated workers, and sophisticated technological and financial support. These things were concentrated in Northern manufacturing and services. Slavery was, in this view, not necessary to the emergence of the US as an economic superpower in the 20th century.

You're right it's broad and hard to answer in one swoop. I think it's useful to break it down into what I see as the key issues. It's also useful to draw a distinction between indentured servitude, which allowed Europeans to purchase their freedom after a finite amount of time, and slavery, where bondage was usually permanent.

One question is how essential slavery was during the colonial/Early American period. The answer is: a lot, especially after Bacon's Rebellion and the decline of indentured servitude. Slavery was a labor force there to replace it. You can read about this in Edmund Morgan's book American Slavery, American Freedom.

Another question is how central slavery was to what we broadly refer to as "industrialization," which in turn was critical to propelling the United States into "superpower" status. There is an increasing body of work that shows how contemporary accounting, management techniques, banking, and insurance grew out of the slave trade, to say nothing of cotton production itself, which had both multiplier effects (textile sector) and comprised a huge percentage of US exports in the antebellum era. The 'sectional' interpretation of the United States, which draws a sharp line between Northern and Southern states during this period, belies how interconnected they were. Many merchants in the North made a lot of money off cotton and other goods produced by slave labor.

A third related -- but ultimately distinct -- question is whether slavery had to have existed in order for capitalism or the United States's rise today. I'm not a huge fan of this question because it's ultimately a counterfactual. Maybe we could've seen a similar rise without slave labor, maybe not, but we have one reality -- what happened -- and slave labor was central to it.

There is an older debate, rooted in questions about the origin of capitalism, where some scholars argued the basic forms of capitalism: paying people a wage for their labor, divisions of labor, technological innovation in pursuit of maximizing profit, and so forth, developed out of Early Modern agriculture in Great Britain, then spread. I think you can take that account, which I basically agree with, and hold it in your head at the same time as you acknowledge the centrality of slavery.