No. Setting aside the question of whether today's conflicts should be understood primarily in religious terms, medieval events do not directly determine phenomena today. At least not in the way that modern historians understand causality, and here I'm deferring to the discussion of causality written down by the great historian Marc Bloch, just before he was executed by Nazis in 1944.

Bloch discriminates between causes and conditions. Conditions, Bloch says, have a degree of permanence. Could a man fall from a mountain side without the geologic forces that thrust the mountains upward? Or could he fall without the gravity which drags him to the ground? Absolutely not, but it would be foolish to think that either of these actually caused the man's death. Geology and gravity were just as necessary for him to climb up the mountain as they for him to fall from its side. These things constitute the conditions which determine why an event turns out the way it does, such as why the man landed where he did or why the ground broke his body with such force, but they don't cause the event itself.

Now, Bloch admits that distinguishing conditions from causes is a bit tricky—he actually calls it arbitrary—but he notes that historians have habits that help them choose between the two. Causes, he says, are often the most immediate—lawyers might say proximal—antecedents to an event in question. They are usually the least permanent or enduring of factors. They tend to be particular things. And they tend to draw our attention especially if they involve people. So it was not gravity or the geology that caused the man to fall, but rather the man's ill-advised scramble up a slippery rock face.

To return to the specifics of your question, then, we can ask whether medieval wars have always caused conflicts between Christianity and Islam. If the answer is no (and it certainly is not), then we should consider these things to be conditions rather than causes. Medieval conflicts help give us the language of jihad and crusade, they offer us potent imagery of noble pilgrims and seething Arab hoards, and they give us historical grounds to justify the actions we want to take today.

But the siege of Acre has as much to do with current events as does the rape of Constantinople. It's not the medieval events themselves that have historical causality. It's our choice to appeal to these events, to interpret them in particular ways, and to give them force and meaning in current events. But the past itself is dead and gone.

As a quick addendum: Why, then, study the middle ages, or any other period that does not directly determine events today? I imagine that there's probably more answers than there are historians' elbows, but to offer just one, it was Marc Bloch's ruminations on medieval history that helped him develop the theories of causality that I've used to try to answer your question today. The past gives us a way to consider basic human problems, and it gives us a vocabulary to communicate with. I think it's one of the most potent tools that the human consciousness of time has put at our disposal, and judging by your question, I suspect that you see at least some of the same qualities in its uses today. Thanks for a good thinking question!