The word is not exactly steeped in tradition, as it only began to gain in popularity after the Vietnam War. I served in the USN until 1975, and I never heard it until I saw some movie about the invasion of Grenada. Leathernecks who served during the Vietnam era have commented the same.
The official website for the Marine Corps (USMC) admits it is only of relatively recent origin, and gives the followng explanation:
Marines and historians have determined the true origins of "Oorah" lie with recon Marines stationed in Korea in 1953. During this time, reconnaissance Marines in the 1st Amphibious Reconnaissance Co., found themselves traveling via submarine to where they were needed. The memorable call of "dive, dive!" would be called on the intercom and a klaxon alarm, which made a very distinct "Aarugha" sound, would announce the descent of the sub below water.
As the Marine Corps Training Reference Manual on the history of Marine Recon is named “Aarugha!”(1989), the explanation that it owes its origin to the sound of the Federal Signal H-8 Klaxon gains in credence.
So how did "Aarugha" become part of the common lore?
From Gunny G´s Marines site (concerning the origin of the "Oorah" cry):
Since May I have been attempting to contact Major Marte for his verification of this story. On 12 August 2002, I received the following e-mail from the major.
When I was in ist Amphib Recon Company (54-57) when we went on our conditioning runs we would chant and one of the sayings was "AARUGHA" which was imitating the sound of the klaxon horn on board the submarine whenever the announcement was made "DIVE, DIVE". This was started by SgtMaj Dave Kendricks (Then a Gunny in 1952 in Amphib Recon) Today, it is part of the Marine Corps language as is Semper Fi, Gung Ho, etc. Loosely translated it means acknowledgement to a question and anything positive. Hope this helps!
Gary "Buddha" Marte
Major, USMC, Ret
So what does "Aarugha" have to do with "Oorah"?
The official USMC website continues...
...recon Marines, who heard this sound often, started using it as a motivational tool during runs and physical training. Over time, the word "Aarugha" came to be too much of a mouthful, and eventually molded itself into the familiar "Oorah,"...
Anyone who has ever marched in cadence or gone on a chilly mountain run can understand how this might happen.
In addition, other US Military branches have some similar form of the cry, although the USMC will deny any connection between them.
US Airborne (WWII)—“HOOA” which meant “Head Out Of Ass” and thought to derive from the radio operators HUA, “Heard, Understood, and Acknowledged”.
"Hooah" is heard in the US Army to this day, and generally understood to mean basically about anything but "no".
As a sidenote, some military stategists are lamenting the fact that due to the land-locked nature of recent conflicts, the use of Marines in sub-based amphibious assault operations has fallen out of practice, leading to the question whether a battle cry originating in the sound of the “Silent Service´s” dive horn is still valid.
There is no way to make an etymological connection between Hurrah, pronounced Hoo RAH, and Oorah, pronounced OO rah. In light of the fact that there is no evidence to support this idea, it must be assumed there is no connection. It is impossible to prove a negative: thinking otherwise is a fallacy.
However, I will take a shot at it. I tried to go to the heart of the OPs question by finding the accepted version; now I will have to find reasons why there is no demonstrable connection (an almost impossible task).
Hurrah, with the stress on the second sylable, is listed in Etyonline with this:
1680s, apparently an alteration of huzza; it is similar to shouts recorded in German, Danish, and Swedish; perhaps it was picked up by the English soldiery during the Thirty Years' War. Hurra was said to be the battle-cry of Prussian soldiers during the War of Liberation (1812-13), "and has since been a favourite cry of soldiers and sailors, and of exultation" [OED]. Hooray is its popular form and is almost as old. Also hurray (1780); hurroo (1824); hoorah (1798).
Although I am sure American soldiers used this exclamation during the Cival War, Spanish American War, and probably up through WWI, probably the last time I can recall hearing this actually being verbalized was a in A Christmas Carol (1938).
Oorah, with the accent on the first sylable, did not come into general usage in the US Marine Corps until well after the Vietnam War.
Please see the following comments by other Vietnam Era Marine veterans concerning the use of the cry:
I'm a 70 year old marine, in watching programs dedicated to the Marine Corps,
my friends and i have a question. When did the Marine Corps decide to adopt the army's uh-ra [sic] and quit teaching and using GUNG_HO. it's a damn shame !!!
Thank you. Being a 1966 to 1970 active duty Marine I have wondered how that worked its way in to the lexicon. I agree with ballmagic--I like gung ho.
Now the question would seem to be, did the USMC actually adopt the US Army "Hooah"-understood to mean "Heard, Understood, and Ackowledged" (HUA)?
There is no anecdotal or historical basis for this.
Instead, we have a cry which, according to admittedly anecdotal sources, is a morph of an onomatopoeic chant derived from the sound of the H-8 dive horn, a sound usually transcribed as Ah OOH gah, or sometimes Ah ROO gah. I have found nothing to contradict these sources.
And there is this from the Marine Times:
Marines, of course, would never say “Hoorah!” They also wouldn’t say “Hooah!”, which is an Army term. Marines say “Oorah!” and are quick to point it out to anyone who gets it wrong.