Nope. The 23rd Amendment has nothing to do with it. The establishment and definition of DC is part of the original Constitution, and it is, by definition, NOT a state. And I see no way it can legitimately be considered a state unless an amendment is passed to allow it.
"The Congress shall have Power To… exercise exclusive Legislation in all Cases whatsoever, over such District (not exceeding ten Miles square) as may, by Cession of particular States, and the Acceptance of Congress, become the Seat of the Government of the United States… (The U.S. Constitution, Article I, Section 8, Clause 17)"
Ok. I was looking in the ammendments and forgot about that establishment in the meat of the Constitution itself. Yeah, with that in there it’s pretty hard to make it a state without ammendment. I stand corrected.