Can a king be checked with another king when checking square is protected by the side which gives the check? Can a white King move next to a black King to check it, if there is a white pawn protecting the white King, preventing the black King from taking the white King?
Hi David and welcome to chess.stackexchange.com. This is a common question. 

It's illegal to move a king into check, or to leave it in check. This includes moving a king adjacent to the opposing king. If you have no legal moves then the game is over: if your king is in check then you are checkmated and you have lost, otherwise you are stalemated and the game is drawn. This is basic to chess, and doesn't require any particular Laws.

If you want chapter and verse, you can find it in the FIDE Laws (which operate in all parts of the world apart from USA) Articles 1.4.1 &amp; 3.9.2. While reading Article 1.4.1, you might also notice that it's illegal to capture the opponent's king. You might reflect that this is a curious rule in two respects (a) it can never apply in a legal position - because it would have been illegal for the opponent to have left his king in check! (b) it would never be a good move to capture the opponent's king - because then you could never checkmate it! You would be right in this reflection, but the rule is there to stop things get too crazy if an illegal move isn't spotted immediately.

To summarize: you can never move so that your king is in check, even if the king is "protected".