http://www.firstworldwar.com/source/espionageact.htm
He'd probably be charged under Section (1)(b)
(b) whoever for the purpose aforesaid, and with like intent or reason to believe, copies, takes, makes, or obtains, or attempts, or induces or aids another to copy, take, make, or obtain, any sketch, photograph, photographic negative, blue print, plan, map, model, instrument, appliance, document, writing or note of anything connected with the national defence;
And U.S. courts have allowed U.S. laws to be applied to acts outside the U.S. For instance if you engaged in "sex tourism" and had sex with a minor you could be prosecuted.
That said, if I was defending him against the charge I would start here:
(a) whoever, for the purpose of obtaining information respecting the national defence with intent or reason to believe that the information to be obtained is to be used to the injury of the United States....
Nobody has alleged any real injury nor an intent to injure the U.S. Assange's stated intent is to
help the U.S. by forcing the government to be more honest. Now the prosecution will argue that the leaks hurt the U.S. by crippling diplomatic efforts. But the burden of proof is on them. And it's difficult with the Obama administration running around doing damage control and saying this is all "no big deal". Every leak so far has either been innocuous (Putin is a mobster), in line with the Obama/neocon agenda (the Arabs hate Iran too) or exposing criminality (Hillary steals credit cards, DynCorp rapes children). In a fair trial Assange could actually win. That's why he won't get a fair trial.