We can never know for sure how things would have gone if something in the past were different because it didn't fall out that way, but there's no reason to believe that American slavery would have died off without outside intervention and just about every reason to think instead that it would have continued indefinitely.

That said, there used to be a historical argument to the effect that slavery was doomed due to its inherent inefficiencies and limited ability to change. The notion was advanced by historians of the blundering generation/needless war school of thought, popular between the World Wars and with some lingering holdouts into the Sixties. They had a bunch of terrible ideas and little of their work is accepted today, but the one that matters for our purposes is that there was a natural frontier for slavery. The climate and terrain would limit the expansion of plantation agriculture and that would be it. Unable to expand, slavery would slowly collapse.

The argument has two big premises, both of which are false. That plantation agriculture is the only viable vehicle for slavery is something that makes sense at first blush. It's literally where most enslaved people were tortured for profit and if we look at where slavery succeeded going way back into the Colonial era, it's largely in places where you could have rather large farm holdings without too much trouble. Rocky New England is not that, but Upstate New York and Pennsylvania were. Likewise, of course, the South. The thing of it is that enslaved labor ended up used for far more than large-scale agriculture. Enslaved people worked in skilled trades, as domestic servants, as urban laborers, and even in reasonably modern factory settings like the Tredegar Ironworks. Those options remain available for slavery to spread via even if cotton cultivation peters out somewhere in Texas...which we also know it does not because subsequently cotton has been grown all the way out in California. Furthermore, even at the time enslavers were looking west with an eye toward enslaving people to work in mines.

Which brings us to the second premise: that if slavery is contained, it must decline and die. This is a belief shared by most white Americans of the era, though not all of them. It makes intuitive sense that in a growing country, something that's not growing is in either relative or absolute decline. That slavery is static doesn't mean it's really going away, though. As long as slavery is profitable -and it's relatively cheap, in money terms- it's not going away. Even excess enslaved people beyond one's designs of avarice wouldn't do it because slavery is an economic system, but also a moral one. By that I don't mean slavery is moral in the sense that it's good and to be admired, but rather it carried with it a moral code with its own ideas about power, right conduct, and so forth. Believing African-Americans who lived among them to be members of a degenerate, savage, barbarous, or otherwise monstrous and inferior race, whites are not content to live with them under the same rules or any kind of level playing field. Giving the liberties of a white man, black men would, so they thought, engage in monstrous campaigns of rape and violence that would necessitate a genocidal race war in order for whites to save themselves. Slavery was the only thing that could prevent that, to their minds. If it cost them a lot of money to prevent that, so be it.

Once you eliminate those considerations, there's not really a path toward ending slavery that doesn't involve some kind of state power being used to force the issue. At that point, it's hard to argue that slavery ends on its own. Rather, someone ended it by means of a power the enslavers were unable to resist. That puts us right back to antislavery Americans, Lincoln, and Civil War.

Probably slavery would have ended at some later point had the South not seceded and attacked Fort Sumter.

But the idea that slavery would have "died out on its own" seems unlikely. The number of slaves in America was growing steadily and pro-slave legislators were fighting hard to expand slavery to new states joining the USA. Many southern states had laws that forbid manumission (freeing your slaves). So if slavery ended it would almost certainly have been imposed on the south by legislation or force of arms.

Doris Kearns Goodwin in 'Team of rivals' does a good job showing that it wasn't merely the threat of ending slavery that drove secession, but the threat of adding free-states and at some point in the future having an abolitionist majority in the senate. The mindset of the pro-slave bloc was very much "slavery now, slavery forever."

Not paying people for their work is always a good way to increase profits, which is why slavery persists in the modern world in so many places despite its illegality.

Western countries all eventually abolished slavery, so perhaps the US would eventually have followed suit but this takes us entirely into the realm of speculation.

"Bleeding Kansas" offers some indication that both sides saw increasingly saw violence as the solution in the lead-up to the civil war, so war was perhaps inevitable even had Lincoln not been elected.

But if abolitionism didn't exist (which again takes us into not-very-useful speculation) it seems more likely than not slavery would have endured, as the structure of the federal government (specifically the Senate) gave Southern legislators disproportionate power to block legislation, which they did on civil rights issues for decades.

I think you are right that this argument in one employed by Confederate apologists, the idea that war could have been avoided and slavery ended without violence at some unspecified later date seems to implicitly lament the war as a greater tragedy than slavery, which in terms of empirical human devastation and suffering is quite false I think.