Can two Kings stand next to each other? 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.