If you take the “original” ship and scrap it, then the guy who invents the time machine in the future won’t be able to go back in time to become his own grandfather.
Could demil it and sell it to Elon Musk, not only would it be more stable, but with twin reactors you could make all the LOX you wanted.
Problem there, is that if you try to scrap it, the universe may get annoyed with your attempt at making a paradox, and find a way to keep you from following through on your plans.
Oh, yeah? Where you gonna get the bagels, smart guy?
Or you could take it as a message from a parallel universe where the Good Guys are losing the war (battle damage was mentioned), and have sent some physical evidence through the Portal as a plea for assistance from their cousins.
Beat me to it. The CT crowd will be all over this thing like stink on s**t.
The sheer volume of highly pressurized crazy spewing out will probably shut down the internet, if not the country. Not to mention all the instant prophets of doom profiteering like mad off the gullible.
I immediately thought, “Philadelphia Experiment in reverse.”
So find out who is mucking about with secret time-travel experiments…
–G!
ETA: Wow! Crowd-ninja’d! 
Thanks for the answers everyone, some really interesting ideas!
Most people are assuming its a result of time-travel but this is closer to the scenario I was personally thinking of, but of course how could you tell? 
Cruise ship! Think of all the tennis courts!
Since the OP’s topic has run its course, time for a related question: what is the US Navy procedure if it captures an enemy warship? It would take far too long to train USN personnel to man it, and the warship would probably be next to useless, so…just tow it home as a museum piece?
I like the idea of demilitarizing it, and selling it to some billionaire; let them figure out what to do with it.
That is what they did with U-505, now in Chicago
Study it if it has any useful technologies, code encryption, or anything like that, then salvage it for scrap, or scuttle it and turn it into an artificial reef.
Unless we are particularly hard up for military equipment, I don’t see us using captured vessels.
The USN commander almost got court-martialed for it, since it could have let Germany know their codes had been captured.
In a real war, the procedures would likely be defined ad hoc.
It would be studied and dissected a LOT.
Some further thoughts:
(1) “Hoax” isn’t really a possibility once you have done even the most basic survey to determine that it is in fact a full sized, “real”, aircraft carrier. It’s not like I could just go buy a spare aircraft carrier and paint “USS Gerald Ford” on it just as a prank.
(2) If it appears to have been occupied, you could learn a lot by looking through the personal effects in the living quarters. Are there any personal effects that identify, by name, any of the crewmembers? Hair or nail clippings or anything that might contain DNA? (This should probably be all done in as quarantined a fashion as possible.)
(3) Assuming that no mundane answer is presented, we’re fairly quickly into the realm of science fiction, at which point, it’s pure speculation what happens next. I’ve read a lot of fiction in which situations like this lead to the government assembling a panel of experts, including military people and scientists but also futurists, science fiction writers, and others who would have spent a lot of time randomly thinking about things like this. (One of them would probably be played by Morgan Freeman. Also, Amy Adams.). My guess is that the first question they would try to answer would be whether this was an alternate timeline Gerald Ford or a from-the-future Gerald Ford. I bet that you could learn a lot from decay patterns on various electronic components. Or, as someone suggested upthread, something involving decay of the fuel rods. Heck, there are probably plenty of fairly mundane components which are serviced with some frequency, where a close inspection could determine how much they had been serviced, etc.
(4) Another interesting question is the extent to which this is the “same” as our Gerald Ford… pick some random component, like the third tack holding the carpet down in the captain’s cabin, pull it out, and compare it under a microscope to the one from our Ford.
Not quite sure where you go from there, but as others have said, the economic, scientific, theological and military implications of the fact that the ship exists and arrived DWARF whatever benefit might accrue from having one additional aircraft carrier.