According to the Qualifications of Office for Secretary-General of the United Nations, no citizen of the ''Permanent 5'' can be nominated for or elected to the office of Secretary-General as it would cause a power imbalance.
I haven't been able to find a part in the series where they mention this issue nor have I seen any discussion about it in YJ fan communities. Is there a logical reason for how Lex was able to achieve that position despite it being illegal for him to hold it?
Luthor's rise to the Secretary Generalship seems to be largely based on the perception that he just saved the entire world. With that in mind, and given the disastrous premiership of Secretary General Tseng, it's likely that Luthor's campaign resulted in a change to the rules/precedents governing who can be the Secretary General.
For the record, a simple majority vote with no objections is all that's needed to make someone the Secretary General and while there appear to be unwritten conventions that the role shouldn't go to a citizen of one of the Permanent Members of the Security Council, this doesn't appear to be a codified rule of the UN, and even if it were, a simple majority in council (with no vetoes) would change that.
"Sounds like someone's finally listening to old G. Gordon, The lone voice of reason, calling for the ouster of U.N. Secretary General Tseng after his disastrous collaboration with the Reach. Well, guess what, folks. Tseng finally resigned. You're welcome. Now, his successor has not yet been chosen, but I think we all know only one man qualifies. The man responsible for saving all our lives from the Reach's MFD's, Ladies and gentlemen, I give you Lex Luthor!"
Young Justice: Endgame
This, of course, assumes that such a rule even exists in the Earth-16 universe in the first place.
