When Alexander the Great died they asked who his empire should go to because he had no heirs he said "To the strongest" and his four generals divided up the empire and fought to each ones destruction. Now switch that four with 86 and you'd guess the level of destruction.
I agree with the previous entry which stated that the Legion will fall apart, falling into weaker internal divisions. The problem I think is that those divisions will splinter a lot in a very short period of time, with each "Legion faction" becoming little bigger than tribes; an ironic yet perverse return to their roots. I would speculate the reason would likely be due to relatively inadequate infrastructure: Considering the Legion grew from nothing in the space of a few decades it is likely it devoted all its economy and effort to expansion (at the cost of civic construction and infrastructure creation) as well as having a relatively sparse population (though organised now its pure-tribal pre-history would have created smaller population density.) If I'm correct about the above most of the Legion successors will "inherit" an "empire" of mostly empty space and pre-war rubble, populated mostly with small towns and slave populations who would try their hand at insurrections once in a while. Compare that with the Diadochi after Alexander who each inherited wealthy rural and urban regions. Most Legion successor nations will be trying to best the others by raiding the factions with more or seeking to restore the Empire by seeking to absorb the weaker ones. So its possible in 30 years they will be tribes again or if they are lucky post-RE warring European nations who could perhaps build a future, perhaps not (though nothing compared to the strength of independent Vegas or the NCR.)
The game is quite consistent in what will become of the Legion. Mr House predicts (if you kill Caesar) that although the Legion will stay temporarily strong for the Hoover Dam Battle, it will deify Caesar and start to fall apart within a year. Joshua Graham (previous Legate) is certain that the Legion will crumble (if you tell him Caesar's dead) and hopes that Arizona's people is spared the worst of the collapse, a man with insight indeed (though not necessarily objectivity.)