The confusion here (online) is that in the move preferences, you can set it to touch move. But as clarified by you all, that isn't the same touch-move that is played in a physical game of chess. Here, online, you can select a piece by highlighting it, but you aren't confined to moving that piece. It's two different worlds.
I was taught that you must always say check when you have your opponent in check. It may be another rule that isn't fixed.
Daniel
well, the rule "touch move" just means "you must move that piece if it's legal to move it somewhere", not "you must make that exact move if it's legal and if it isnt then we need another rule to ensure you must still move that piece if it's legal" one rule covers the lot.
The check thing is just a courtesy. For example you wont lose a game for not announcing check, and also it has no affect on the touch move rules. You just wont feel so bad when enforcing the touch move if you have announced check 
The setting on chess.com i believe just removes the "submit" button and makes the move automatically when it is completed. You can still deselect pieces once you have clicked them, as it is all too easy to accidentally click the mouse, compared to accidentally touching a piece in real chess. I know it mentions touch move in the settings to do with whether or not you are able to observe your piece in its new location, but this is not technically correct, and seems to be the source of the confusion.
you aren't required to say check, it is the player's responsability to notice check