The screenshot below shows the result of a demand I made on an AI civ after they attacked one of my units.  The response came several turns after the initial demand, but I've had it happen immediately upon the demand being made.  The Game (not the AI CIV) is telling me, the human, that the ONLY choice I have is to withdraw to demand or declare war - as in I will be the aggressor in the war and my war support will go down each turn if I do not prosecute the war and opt to do nothing.  This seems patently unfair.  Should it not be the AI Civ telling me to withdraw the demand or THEY will declare war?  A frustrating situation unless I specifically want to go to war.

They decline your demand. So you are presented with a choice. It doesn't make much sense if your demand still stands because they already said they have no intention to accept it.

You can do the same thing to the AI. Click "Refuse" and the AI has to either withdraw the demand or go to war.

It has to be that way because otherwise the AI could decide that your fondness for tomatoes bugs it and it has a "tomato policy difference" grievance. If you had no way to force the issue, then you'd be blocked from trading with that AI for potentially the rest of the game over savory fruit preference. Because you can force the AI to war or withdraw, it can do the same to you.

Note, if you don't want war, just renounce up front. You get a slight diplomacy boost with the AI every time you do so.

If you are trying to build war support, just go ahead and click withdraw. Your avatar will say something meek and embarrassing, but your civilizations war support will go up.

The nature of demands basically is that you should only make them if you intend on going to war, or expect the AI to declare war on you. Either you get to have your demand on the table for when war is declared, giving you a guaranteed benefit if you win, or you get more war support if you withdraw. 

Although maybe a small, but significant, influence cost tied to refusing a demand would be okay. It might allow demands a few turns to "simmer" and rack up war support for the offended party even if the offender is ready for war.

Yep - you are all correct - the more I think about it, it's the only way it could be.  Also I am finally starting to notice situations where the animated/voiced dialogue of a leader doesn't necessarily match what the diplomatic screen is saying - and go with the latter.  Thanks for the quick feedback as always!