Monty made the same demand for Education, but I did not give it. It seems that not giving away techs is my problem. It already caused 2 wars against me (Pacal in the 400 AD save and this 1200 AD save).
I thought switching to Free Religion would please Monty, but I think it doesn't work that way.
Up to a point, Free Religion could help--it would probably be sufficient to bring Monte to Cautious from Annoyed. However, that situation would just REDUCE the chance of you being the war target. Monte will still "roll the dice" on each turn to see if he wanted to go to war against you--he'd just be slightly less willing to do so.
Being Annoyed with you just means that if Monte wants to go to war with you, he'll choose to do so. If he's Cautious with you, there is a 30% chance that he'll let bygones be bygones and will decide not to attack you afterall, even if the "dice roll" said "attack that Washington jerk."
If you get Monte to Friendly, then he won't be able to attack you at all.
However, when you refuse a Demand, many AIs will start to plan a "Sneak Attack." A couple of "honourable" AIs will declare immediately, but basically, if you refuse a Demand and you don't have insanely big Power, you should expect to be attacked.
You won't always get attacked, but if you PLAN for such an occurrence, you'll at least be ready for it when it does happen.
To me, if an AI is not the tech leader, I tend to give into their Demands. It's a great way to get free positive Diplo modifiers, although you should have an idea of what the Worst Enemy situation is like--giving into the Demands of someone else's Worst Enemy could anger the person who dislikes the AI that made the Demand in the case where you give into the Demand.
In that situation, you have to weigh the risk of being attacked immediately or in the very near future by the AI making the Demand versus angering the AI that really hates the AI making the Demand.
Also, consider that if you don't want Monte to attack you, you want to INCREASE your relationship with him, instead of souring it. Refusing his Demand makes the relationship go relatively 2 points in the wrong direction (you missed out on the You Gave us Tribute or the You Gave us Help +1 positive Diplo modifier and you also incurred a -1 negative Diplo modifier for Refusing the Demand).
So, if you go around Refusing Demands and Refusing Requests for Help, then you're not really on the right path towards getting said AI to a Friendly status.
In this game, the decision was easy--Monte is behind technologically so he has nothing to offer in trade. Thus, we won't "lose out" on a potential trade by giving him a tech "for free." However, remember that it is NOT "for free." He actually GIVES you a +1 Diplo modifier and DOES NOT GIVE you a -1 Diplo modifier, which is essentially an easy way for you to be +2 Diplo modifiers ahead of the player that refused the Demand.
Sometimes I will refuse a tech Demand from the tech leader if I want to trade them that tech within a couple of turns, since by the very fact that they are the tech leader, they are probably focusing more on their economy and less on a military and thus are less likely to attack you. Trading with them shortly thereafter will also tend to make them like you a bit more (either a potential increase in relations or else a temporary hidden value of "you traded with us within the last 5 turns, so we're going to ignore that 'dice roll' that told us to make a Demand from you"). Note that that last bit about a recent trade is more anecdotal than code-digging based info--there IS a value in the code that "remembers" a recent tech trade, but I am just using anecdotal evidence in my claim as to how that value actually gets applied.