It does not matter AT ALL! The EOD from my class (2010) were from all divisions. Do well on the screener and the interview - even class standing as long as it is top half will not hurt you. One of the top in our class, top 10 and a Div 1 major (Engineering), did not get their first choice of EOD. Do not pick a major just because you think it will "help" you get your service selection - it does not work that way. Screener's and interviews are actually more important.
True, your major does not matter
unless you want to go into the Medical Corps - because, unless you take the courses that are emphasized on the MCAT (Medical College Admission Test), you will not do well enough to even be admitted into medical school - in which case, the Naval Academy will have little choice but to DENY your Medical Corps selection - regardless of QPR, class standing, or any other standard.
If all the medical schools to which you applied do not think you're good enough (because of unimpressive MCAT scores), it is very likely that the Naval Academy will agree.
Generally, this involves majoring in Chemistry. However, if you have validated many courses, you may have the latitude to major in something
other than Chemistry, as long as you have room for the Organic Chemistry and Biology courses in your schedule. There are those who have gone into the Medical Corps
without having majored in Chemistry - but it is somewhat rare. There are really just a handful of courses that are needed to properly prepare for the MCAT, some which are
required at the academy even if you're a non-technical major (i.e Calculus, Physics, and Chemistry) Plus, there are parts of the MCAT that involve reading and non-technical interpretation.
So, I guess, even in the most extreme case (wanting to go into the Medical Corps), your major does not really matter all that much. I do not think there is any service selection that is as selective.
In a past admission catalog it was stated that there was room for up to 25 slots for the Medical Corps. That has not been repeated in subsequent catalogs. And, the number going into the Medical Corps have been far below that.
Which is odd, because one of the academy's recruiting videos, in the "Fulfill your Destiny" series, is 100% dedicated to promote the notion that a candidate can go to the academy and become a doctor.
Complete propaganda - and borderline false advertising.
Less than 1% of the midshipmen went into the Medical Corps from the Class of 2010 - hardly worthy of a video.
http://www.usna.edu/fyd/index.php?video=ramchandar