A friend of mine is in the Navy, worked in supply on ship including an aircraft carrier. He claims he "doubled as a corpsman" when the "marines went ashore".

I find this hard to believe. A Navy corpsman is not something you can take a four class on and then just do, no less would you be deployed with Marines into a potential combat zone.

He is full of shit isn't he?

Maybe he did so in some kind of support role? I don't think you can serve in two branches at the same time... Just my ignorant guess..

As a  former Sgt. in the Marine Corps let me try and give you some insight about Navy Corpsman.

The Marine Corps does not have Marines as Corpsman.
All Corpsman attached to the Marines are members of the U.S. Navy and they are VERY well educated and trained.
Looking back as best as I can remember, The corpsman I remember were in 2nd and 3rd enlistments.
You didn't clarify your friends length's of enlistment.
I might question, unless he had some formal schooling or training prior to his enlistment in the Navy, or might have some doubts if this is his first tour of duty.
I don't recommend you calling him a bullshit artist, especially if he is still a Marine Corps Corpsman in front of other Marines....LOL

One of the fastest ways I know for a person to get the living shit kicked out of them is to insult or mess with a Navy corpsman with the Marines.
Marines have the utmost respect for there corpsman. They walk on water in the corps.
You See,... Navy corpsman assigned to the Marines,.. Have a habit of not living long  being assigned to Marine units in combat.
Some of the bravest men in combat.. I wouldn't even think of trading jobs with a Navy corpsman...NO WAY

Hope that helps.

You can have Primary MOS's(military occupational specialties) and secondary ones and small billets which is how they probably ended in Supply to fill a slot. 

As a Sergeant, my primary MOS was Radio Repairman, but I was a Range Coach, MIMMS clerk and held billets as NCOIC, Publications NCO, Tool NCO, Calibration NCO, Combat Conditioning Instructor,  Substance Abuse Control Officer, Platoon Sergeant, Section Head, etc.

Heck at some point I was in Shipping and Receiving, Company Office and worked in Battalion Headquarters.

Sometimes they send people to fill open slots or because someone in charge thinks their competent enough to have a job that's not directly related to what they normally do.  Also after your first two years of enlistment you can lateral(lat) move to a different Primary MOS if you're lucky or Lat move when you re-enlist for another term.  So long story short, odds are he did both.
I knew a Captain in the Navy who went from being the head of Mainside Dental on Camp Lejeune to become a Navy Chaplain and also a Corpsman who just guarded the Chaplain of our battalion when we did a 8 month stint in Afghan. 

Hey Marine were not talking about a jack of all trades ...Master of none.
All Marines are basic 0300 Infantry from the commandant of the Corps down, Every Marine from the Commandant down every year has to qualify on the rifle range with the basic infantry weapon.
No way in hell is a sailor going to be given a Corpsman med pack and sent in as a Corpsman with a NEW FIRST AID CARD with a Marine platoon.
Your describing non critical MOS personnel filling non critical MOS JOBS.

Give me a break, Your going to have these people believing the Marine Corps operates like a bunch of hay seeds.
                                                                  Your a little lose there Marine...
                                                                                               
                                                                                                            LETS
                                                                                     TIGHTEN UP SARGENT..!!

This is the Marine Corps I remember, 3min.video of the movie " THE D.I. " as close to the real thing as the corps would allow to be shown.

                                                                              https://www.youtube.com/watch?v=irCqA4lcgoE