Oh the Mongols who remained as a unified entity for a total of twenty-five years?
Their successor states lasted much longer, as would Legion successor states, but the Legion as a single nation would not last long past Caesar's death.
If it didn't collapse into different nations like the Mongols, it would likely collapse in the same fashion as the Galactic Empire in Isaac Asimov's Foundation Series.
For those who are unfamiliar, the Galactic Empire remained together as a single nation but it slowly apart between planets declaring independence and powerful generals seizing the imperial throne time after time. This death would be a lot slower and more draining.
I'd personally like to see the Legion slowly turn into a collection of independent Latin-speaking Roman God worshiping nations that, at the very least, sign a truce with the NCR and Vegas (if it's independent) to make sure that none of these factions conquer another.
All of my previous thoughts and ideas for the end of the Legion were inspired by the Roman Republic or the Roman Empire but perhaps something along the lines of the dark ages would be a more fitting end. By the I mean the Great Khans.
The Roman Empire's consistent enemy was the Germanic tribes even though the Romans far outmatched the Germans in nearly every way. When the Romans were at their weakest the Germans struck and destroyed the Roman Empire, setting off the dark ages.
What if the Great Khans are being set up to do the same. If you convince them to leave instead of suicide bombing Hoover Dam, they can go off to the northwest to forge a mighty empire (I know I mention this a lot but it's just intriguing to think about how an empire can form or the people who would go on to form it can kill themselves based off of what a single person convinces them to do). What if this is so that when the Legion collapses into civil war, the Great Khans storm in and conquer it. The Khans would hold a grudge against the Legion due to the Legion's plans to conquer and enslave them. This wouldn't be the driving force behind the Khans' attack on the Legion but it would prevent the Legion from trying to become allied with the Khans.
I could easily see Obsidian having played for Caesar's Legion to have gone the way of the Roman Empire and for the Great Khans to be the ones to act as the barbarians.