Nah. People underestimate just how difficult it was for settled communities to muster horse archers. Becoming such a skilled rider that you can shoot while galloping is something you have to spend a lifetime at (which the Mongols did). Hell, even the Seljuks of Anatolia would stop their horses and THEN fire their bows simply becuse how risky it was to do both at once. And those guys where way better horsemen than anything you'd find in Central/Western Europe. No way Europeans would have managed to master such a fighting-style.
This kinda works under the assumption that the Mongols liked fighting pitched battles. They didn't. The Mongols harrased an opponent until death. They feinted retreat and made hit-and-run tactics. They fought wars, not battles.
They managed to pull of such highly complicated strategies becuse of how insanely disciplined and organized they where. You have to understand, things like this simply weren't done in Medieval Warfare. The Europeans and others assumed they where fighting deamons when they saw how swiftly the Mongol armies advanced and the syncronization that they did it with.
The Mongol army was a well-oiled fighting machine, with a clear chain-of-command. Compare this to Europe, where every Nobleman had his own private army (and commanded said private army). It would have been impossible for a bunch of in-fighting noblemen to pull off the sophisticated manuvers necessary to counter the Mongols highly mobile tactics. They simply did not have the military orginazation to do so. The host would begin to fall apart under the stress of the Mongols harrasments, with individual nobels fleeing/defecting/or storming off right into another Mongol trap.
Also, the Mongols where masters of Divide-and-Conquer. They allied with one ruler while toppling another ruler. I do not think that the scattered, belligerat European states would be able to unite against such a force. Pope or no-Pope.
Also... it's not like English longbowmen (the only bow that could out-reach the Mongol bow in Europe) existed in abundance. And training to be an English longbowman was also a lifetime job. Nor would the English monarch's have any real incentive to go off and fight Mongols in Germany or France.