Why can't the king put the enemy king in check if it's protected?

In this position, the king is protected by the rook. King's cannot take protected pieces/make a move that would put them into check. White's king is under no threat as it cannot be taken, therefore it is not in check.

If the black king cannot attack a protected piece(which happens to be the king), why can't the white king go here? 

Sorry if this is a stupid question, I am just interested as to why this isn't the case.

https://www.chess.com/terms/chess-king

"When a king is attacked, it is in check. A king cannot move into check (that is considered an illegal move),"

Which violation happens first?

If you were able to take a protected piece, putting your king in check, why couldn't your opponent do the same?

Pinned pieces still attack squares they could move to if they were not pinned and a king can never move into an attacked square and no piece can be moved if that exposes your king to an attack (check) by a pinned piece.

Thanks for putting up with ours. You'll see ...

because kings cant kill kings

In my college days, we didn't believe in many rules. We used to play without castling and no captures were allowed till the 10th move

Many explanations of the rules of chess are flawed, circularly defining the concepts of "legal move" and "in check" in terms of each other. This leads to confusions like the one highlighted in this thread.

More careful explanations will avoid doing this. There are multiple ways to do this, but probably the easiest to understand is this: First, define the notion of a square being attacked by a piece (which does not depend on check). After that, we can define "in check" to mean, "The king is on a square attacked by an opponent's piece." Then we can state what a legal move is, which will include the check rule: At the conclusion of your move, your king must not be in check.

It occurs for the same reason that checkmate is a win, even if your opponent can do the same on their very next move. You get the king first, which is the point of the game.

Kings can't check kings because a king cannot move into a position where it would attack the opponent's king. This would create an illegal situation where both kings are in check, which is not allowed in chess.

So, according to you, the black king can't move into check, but the white king can?

Very suspicious double standards.

In theory it is debatable only if you use twisted, false logic. You are overcomplicating things and that disguises how simple it is in reality.

And it's really extremely simple. A king can't move into check. That's it, end of story.