Well.... Yes and no?
Slavery was pretty fundamental to the way the americas was colonized, because it was that central and important. The areas that *didn't* rely on either slave plantations or exploitation of large native populations tended to be more afterthoughts, and were never as productive (and even then they often profited from the existence of the slave-trade) It's an immensely complicated thing, lots of capital accumulation wouldn't happen or would happen differently. (even places we wouldn't think of as being part of the slave trade might be eg. building ships, producing food, producing trade goods for barter with africans, etc.)
Could white indentured servants have worked? Maybe, the reason they *weren't* is because african slaves were a "better" option. For starters, european monarchs (or for that matter, african monarchs) didn't much like people taking their tax-paying citizens and sending them to the other side of the world (where taxation and drafting them is much harder...) so that always limited the pool, and that's disregarding the social troubles. The other major slave-market, the black sea/steppe region was both A) getting conquered by the Russians and B) being diverted towards the middle-eastern market (and it was also farther away of course) which meant that getting slaves that way was much, much more expensive.
Europeans also had the nasty problem of tending to get various tropical diseases and die off in pretty staggering numbers whenever they set foot in the tropics. Africans had at least some inherited resistance to some of these diseases.
The third option, native americans, had a combination of these two problems: First of all, they died off in droves from disease. Secondly, the spanish crown tended to at least make noises about people enslaving their theoretical subjects. That didn't stop other forms of forced labour, or even outright enslavement in the conquest-period, but "real" slavery was hard to establish with any permanency. (and of course, even when you did have a large indian population that could be exploited, the spanish would hardly let you ship them to Carolina)
It gets... Tricky, to make alterations to these equations. If you make african slaves prohibitively expensive (say, a few major empires that put a stop to the slave trade arise in Africa) that's going to have all sorts of weird knock-on effects, and I think in order to get a "US South without slavery" you pretty much have to remove the entire trans-atlantic slave trade from the equation, because if there are all those slaves availible *someone* is going to use them in places that are good for growing these specific cash-crops.
Remember, people are thinking of a big influx of european migrants: This didn't really happen. We're not talking about the 19th century with it's exploding population growth, but the 17th century when population is fairly stagnant. That doesen't mean there won't be colonization, but it will look very different and probably at different points.