It is difficult to say, but unlikely. The legendary Courier 6 contributed a number of supporting forces to the Second Battle of Hoover Dam, including two squads of power armored soldiers, a horde of Great Khans, and a B-29.

But of course, the single two contributing factors that the Courier brought to the Second Battle of Hoover Dam was:

• Killing Legate Lanius, Monster of the East.

And

• Bringing a horde of rocket-armed murderous robots to the fight.

We can't really know for sure, but in my opinion they would lose the Second Battle for Hoover Dam.
Without the Coureir's support, the following things will happen;

• Camp McCarren, the NCR's main base in Nevada, will be overrun by the Fiends.
• The Monorail connected it to the Strip will also be destroyed prior to the fight. Furthermore the Omertas will go through with their coup, massacring all of the tourists in the surrounding casinos for Caesar.
• The Great Khans will be used by the Legion to harass NCR supply lines and thin NCR numbers during the battle.
• President Kimball will be assassinated.
• The Misfits will never get into shape.
• Ranger Hanlon will continue to spread misinformation to the Rangers.
• Camp Forlorn Hope will succumb to the Legion, much like Nelson.
• The Boomers will not support the NCR during the Second Battle. Also Lindsay will presumably never be reunited with Mr. Cuddles :(
• The Brotherhood of Steel will not support the NCR during the Second Battle.
• The Enclave Remnants will not support the NCR during the Second Battle.
• Freeside will never receive NCR's relief effort due to high tensions between the Locals and the Squatters.
• The Followers will not support the NCR during the Second Battle, and will most likely be in a rough state when the Freeside rioting pursues.
• The Powder Gangers residing in the NCRCF will be wiped out, however Cooke's gang in Vault 19 will remain in power and the convicts terrorizing Primm will continue to sack the surrounding area. Goodsprings will most likely be turned into a rallying point for the local thugs.
• Mr. House will never obtain the Platinum Chip, meaning his Securitrons will not be upgraded and he will never gain access to his Securitron Army.

Basically, the Legion will knock the NCR out of Nevada and continue to steam roll their way into Vegas, which will most likely already be a mass graveyard. Once Caesar turns it into New Rome and erects his empire he will probably set his sights on California. IMO he will not be able to actually invade the Core Region, which is much more civilized and organized than the NCR's fringe territory, but we really have no way of knowing for sure. All we can guess at is the outcome of the Second Battle for Hoover Dam, which is almost decisively a Legion victory without outside involvement.
Oh, and Caesar's tumor will make the old tyrant kick the bucket soon. This will without a doubt present the Legion with a whole plethora of challenges to overcome in the following years.

Most likely not.

When you look closely at the effects of the NCR and Caesar's Legion expansion into the Mojave, you soon realize that the NCR was grossly over-expanded, and the Legion was a force literally predicated on it's ability to expand and obliterate. The NCR was far from its home provinces, and capitalistic in-fighting had caused its supply lines to almost dry up, with the fewer and fewer recruits that came from the Hub, Shady Sands, and the Boneyard being supplied with even less reliable equipment.

The Legion, on the other hand, had avoided the problems of over expansion that plagued the NCR by creating a monolithic culture centered around the totalitarian figure of "Caesar". I personally think he never existed, the stories of a Follower of the Apocalypse turning into an Iron-Fisted Dictator are absurd, and so are stories of a Pre-War "Roman" empire. We have records that clearly show the only important Pre-War countries were China, the USSR, and the United States. However, this symbol was enough to make the tribes the Legion incorporated fanatically loyal, and the villages captured produced supplies at an alarming rate and with the highest skill.

The matchup here was pretty dire for the NCR, and they knew it. If not for the Courier, the Mojave and even the glorious jewel of New Vegas might just have become another Legion Charnelhouse.