My white King is separated from the Black King by one square (between) which is protected by my white rook. I moved my white King next to the black King putting him in check (from the square protected by my rook). My opponent says that this is an illegal move because two Kings cannot face each other. I contend that my white King can threaten the black King from a protected square (by my rook). Who's right?
Your friend is right. Think of it in terms of capturing the king: check means that your king could be captured on your opponent's next move. If you could move your king next to your opponent's king, your king could be captured on your opponent's next move; the fact that after that you could capture your opponent's king doesn't change that: your king has been captured, and you've lost the game.
Your friend and the existing answer here are both right: You can't do that.
There's no explicit law of chess for just this situation because it's fully covered by a slightly more general article from the FIDE Laws of Chess:
3.9.1 The king is said to be 'in check' if it is attacked by one or more of the opponent's pieces, even if such pieces are constrained from moving to the square occupied by the king because they would then leave or place their own king in check.
This covers both your case -- the square is considered 'attacked' by the opponent's king even though it could not be moved there because of the rook -- and other cases, e.g. a piece pinned against it's king is still considered to attack all the squares it could move to were it not for the pin, hence the opposing king cannot be moved to any of them.
Moving your king next to another player's king is illegal.
However, the USCF rules for blitz chess state
"3b) If an illegal position is created or an illegal move made without the opponent making a claim, the position stands and a claim not allowed when the opponent has determined a next move."
While this provision is obscure, it is sometimes amusing. I remember seeing someone in a clearly won endgame position about to queen a pawn under severe time pressure, while his opponent simply moved his king back and forth. Just before the queening pawn move, the ordinarily loosing opponent checked the guy with his king. The person who queened did not claim a win before pressing his clock, and he was startled to loose to KxK!
However, I see that the USCF has added explicit mention of this procedure to the rule book:
"16.) Moving the King next to another King is an illegal move. Intentionally playing a king next to the opponent’s in order to take the opponent’s king on the next move (if not caught) is a cheap shot and will not be tolerated! Stop the clock and claim a win because of an illegal move."
The rule book does not define the precise meaning of "will not be tolerated!", nor does it mention whether additional sanctions will be made for laughing.
It is illegal to play a move that results in your king being on an attacked square, or, as the official rules put it, emphasis mine:
The king is said to be 'in check' if it is attacked by one or more of the opponent's pieces, even if such pieces are constrained from moving to the square occupied by the king because they would then leave or place their own king in check. No piece can be moved that will either expose the king of the same colour to check or leave that king in check.
Moving your king next to his would leave your king in check (the square is attacked by the opponent's king), so you're not allowed to do that.
"Backing up" a king never makes sense; you aren't even allowed to let your king be captured, so what might happen after that capture is always irrelevant.
I think of it as if your king gets captured, all your pieces are automatically captured also. So if your king is next to his king and your king is protected, if he captures your king your rook is captured too and cannot capture it back. So those types of moves are considered illegal.
Two kings must always be separated by at least 1 square because 2 kings cannot check one another.