As the title implies this post most likely contains spoilers.

Who do you think would win the Second Battle for Hoover Dam without player intervention? We could say that the player either was killed by Benny or just decided that the whole Platinum Chip thing was more trouble than it's worth. I mean honestly if you somehow survived being shot in the head would you chase after the person who just tried to kill you?

Either way the player is now out of the picture. This means that the state of the Mojave will largely be shaped in a slightly more organic way. Being changed by the factions present and their actions as opposed to the whims of an individual.

The Tunnel Snakes would rule the Mojave. Upon migrating west, Butch's charisma alone recruits countless followers. The NCR and Legion wouldn't stand a chance.

the 2nd battle would go to the legion. why, the NCR are having supply issues their morale is already wrecked and the only reson they won the first time was a trick that will not work agian.
however after that the legion will be wasted once the greater NCR sees them as a credable threat

The NCR would appear to win but without the players assistance Forlorn Hope and Camp Golf are wiped out by the Legion during the attack on Hoover Dam. Camp Mccarran is also attacked by fiends who are working with the Legion. Also the NCR was not prepared for the Legion to use the turbines tunnels to gain access effectively slaughtering many NCR troops. Also as erebamagi pointed out the NCR only won due to the Legion being lured into a trap and with the legion learning from their mistakes, this would not happen again. Again as erebamagi sated Cheif Hanlon falsifying intelligence caused unrest with both the troops and higher ups which in turn resulted in under supplied troops and dropped morale. With NCR troops being under supplied as seen at Forlorn Hope legion victory odds are very high especially with the Legions numbers during attacks. Based on all this the NCR while appearing dominant would lose and face ultimate defeat due to their current state without the interference of the player.

The Legion's attacking Hoover Dam from the inside would certainly hurt the NCR, as would a potential successful assassination of Kimball. However, I question whether (in light of Oliver's mass troops at the Dam for the fight philosophy) the Legion has the resources to actually break the NCR's lines (given the superior technology of the NCR). I feel as though on numbers alone against a mostly non-ballistic weapon based opponent, the NCR should hold the Dam.

However, as MrEggs0925 said, the NCR gets badly in other areas during the attack. That being said, the NCR is very big, and very powerful outside of the Mojave, and a second defeat at the Dam breaks the Legion's spirit, before the full NCR war machine travels East and wipes them off the map.

The Legion. As much as I hate to say it.

The real question is how long could they hold it.....

If taking into account what the game presents, then Legion will win the battle but lose the war. Caesar would soon die from his brain tumor and Lanius would push into the West through Long 15. NCR would pull back to defend their borders and Mojave is basically fucked. After that I guess there'd be a stalemate, but at the end NCR would come out on top, but with massive casualties and fringe settlements destroyed. Then Ulysses nukes them all and tunnelers eat the rest.

I feel that the Legion would win the battle, for several reasons. Terrain, leadership, available troops.

The terrain of the Hoover Dam is, for the most part, not defensible. The dam is basically a large concrete bridge with a few spires on it which all lead into the power plant section of the dam which can be broken down as a cathedral ceiling series of caverns. On the western side of the dam is a strategically important ridge.

As stated the exterior has a few towers which can give a select few troops the advantage of having the high ground, but realistically these positions can only temporarily be used as sniping positions before they get overrun. The ridge on the western side at least seems more defensible than the towers on the dam. However Chief Hanlon states that General Lee Oliver positioned the rangers on the front rather than the ridge. It is implied that Oliver did this simply to contradict Hanlon.

The NCR military leadership is seemingly incompetent. Several characters within and outside of the NCR share this opinion and for somewhat good reason. General Lee Oliver's strategy for the battle is garrison as many troops as possible as the dam and then fight the legion head on. Mr. House put it best:
"He wants to out-fight the Legion in a straight-forward slugging match, and then, when they rout, pursue and destroy them in detail."

The legion on the other hand has a semi legitimate strategy. Sneaking in through the intake valves under cover of what appears to be general march on the dam.

The problem with that strategy is that you either need superior numbers or technology for it to work. At least from what we see in game the NCR military does not outnumber the legion, at least not the NCR military stationed at the dam. The NCR has many more troops stationed throughout the mojave, but they are stretched so thin that diverting troops most likely means ceding whatever front they came from. The legion on the other hand has not stretched themselves thin. The bulk of the legion forces you see in the mojave are small raiding groups.

Technology wise the NCR does have a slight edge: their standard trooper is equipped with the equivalent of leather armor and a combat knife, service rifle with at least 2 clips. The legion standard trooper is also equipped with the equivalent of leather armor a machete, 2-5 spears, and healing powder. The troops we see at the battle are not standard though the legion who make up the third or fourth wave are equipped with assault rifles and thermic lances. While the legion does not appear to have as many advanced weapons as the NCR they still do have them. They are simply reserved for their experienced units.

As some have pointed out the legion would likely lose the war.

Without player intervention Caesar would have most likely died from his tumor. While Legate Lanius is more than a capable military leader he is not Caesar and that is an issue that any successor to Caesar would have. I don't think the Legion would collapse after Legate Lanius took control or even after his successor took control. The entire generation of the legion has been conditioned to follow the Caesar unto death. I don't think that would change just because of Lanius. Even is there were an upstart Centurion who tried to rebel, the rebellion would be put down, brutally. Lanius has a record of brutality, his preferred method of punishment being Decimation, having 1/10th of people being killed by the other 9/10ths. Has Caesar himself states the Legion is not sustainable, he never built it to be. Eventually the legion will under go a major societal change. Caesar himself stated that this change would have been caused by the war with the NCR:
"The NCR council will be eradicated, but the new synthesis will change the Legion as well from a basically nomadic army to a standing military force that protects its citizens, and the power of its dictator."

That quote is assuming that the legion win the war. In one of the speech checks that player can pass with Legate Lanius it is implied that as the legion pushes into the NCR. The NCR's weakness will become the Legion's weakness. Unsustainable expansion.