What was the reaction across Christendom to the conquest of Jerusalem and the Holy Land by the Rashidun Caliphate in 637?

The main problem for answering this question is the lack of sources. The Near East in the seventh century is actually very poorly documented, so I don't think there are any sources that explicitly describe the reactions to the capture of Jerusalem. There is even doubt about exactly when Jerusalem was captured: the dating the OP suggests is dependent on the lost Chronicle of Theophilus of Edessa from the eighth century, but the historians who used Theophilus' work (Theophanes Confessor, Michael the Syrian, Chronicle to 1234 etc) all suggested different dates from between 634 and 637. The earliest Arabic sources meanwhile suggest a date between 636 and 638. 637 is very likely, but it is also far from certain, as 635 is another popular choice for some historians. We can make a few inferences about its impact based on the contemporary sources, but it will still unfortunately be a very vague answer.

Realistically, I think there are only two vaguely contemporary accounts of the fall of Jerusalem, one from Pseudo-Sebeos, an Armenian historian writing in the 660s, and the other from Fredegar (a name historians use for an anonymous Frankish historian) in the 650s. Pseudo-Sebeos described the fall of Jerusalem in this way:

They [the Ismaelites] crossed the Jordan and camped at Jericho. Then dread of them fell on all the inhabitants of the land, and they all submitted to them. That night the people of Jerusalem took in flight the Lord’s Cross and all the vessels of the churches of God. Setting sail on the sea in ships, they brought them to the palace of Constantinople. Then, having requested an oath from them, they submitted to them.

Fredegar was similarly brief:

The Saracens proceeded - as was their habit - to lay waste the provinces of the empire that had fallen on them. They were approaching Jerusalem. Heraclius felt himself impotent to resist their assault and in his desolation was a prey to inconsolable grief.

For what it is worth, I'll include the eighth-century historian Theophilus' account as well (this version is preserved in the ninth-century Chronicle of Theophanes Confessor, but other versions are very similar). Theophilus is interesting as he probably used various Muslim sources as well in his work, so we get quite an interesting account that drew on both Christian and Muslim traditions:

‘Umar invaded Palestine and, after investing the holy city for two years, obtained it by agreement; for Sophronius, the bishop of Jerusalem, took a guarantee for the whole of Palestine. ‘Umar entered the holy city dressed in filthy garments of camel hair and, showing a devilish pretence, sought out the Temple of the Jews – the one built by Solomon – that he might make it a place of worship for his own blasphemous religion. Seeing this, Sophronius said: ‘Verily, this is the abomination of desolation standing in a holy place, as has been spoken through the prophet Daniel.’ And with many tears the defender of piety bewailed the Christian people. While ‘Umar was there, the patriarch begged him to receive from him a loin cloth with a garment to put on, but he would not suffer to wear them. At length he persuaded him to put them on until his clothes were washed and then he returned them to Sophronius and put on his own.

All of these accounts are very vague and they don't really tell us that much about what contemporaries thought about the fall of Jerusalem. Still, the fact that these historians noted the capture of the city says a lot about its importance. Fredegar was writing in Burgundy many miles and decades away from the events he described, yet Jerusalem was the only city he mentioned with regards to the Arab Conquests. The removal of the Holy Cross from Jerusalem is also telling, since it had been lost when the Persians captured the city in 614, so its safety was clearly important for the Romans when Palestine was once again under threat. Another useful source supporting this interpretation might be the sermons of Patriarch Sophronius of Jerusalem from this period. Admittedly they were (probably) written before the fall of Jerusalem, but their tone does tell us a lot about his opinions (or at least what he was willing to say to his congregation):

Therefore I summon and urge and beseech you to desire for Christ God, so that, whatever our power, we may reform ourselves and take pride in repentance, and in that atonement purify and curb the flow of deeds that are hateful to God. For if we live thus, as dear and pleasant to God, we may laugh at the fall of the Saracens who oppose us, soon make their ruin feeble, and know their final destruction. For their blood-loving swords will enter into their own hearts, their bow will be shattered [Psalms. 37:15], and their own weapons will ensnare them, and they will provide for us a way free from fears.

Basically, Sophronius saw the Arab invasion as the unfortunate result of human sin. However, it is actually a lot more complicated, since Sophronius was also a Chalcedonian Christian firmly against monotheletism, a doctrine then promoted by Constantinople. Realistically, Sophronius was not only responding to Arab depredations, but also issuing a challenge to the emperor. For him, victory will only come with a return to orthodoxy, whereas further defeats are the inevitable results of heresy. As such, I don't think we can separate the reaction to the Arab Conquests from the doctrinal debates within Christianity, since for these individuals it was one and the same. For these people the fall of Jerusalem was not just the result of military defeat, but also divine intervention in human affairs for their misdeeds, so in a sense the fall of Jerusalem only confirmed their own prejudices and led to outrage against the emperor and not necessarily against the Arabs.

Moreover, we shouldn't emphasise the conquest of Jerusalem too much. Jerusalem had already fallen once before to the Persians and was even subject to a brutal sack in 614 (which admittedly only happened after the locals revolted against the new regime, the initial surrender was entirely peaceful); the first siege was recorded in much more detail by the sources, including a very angry polemic from a monk called Strategos. The fall of Jerusalem in the 630s however did not lead to the same reaction, since as far as we can tell it was handed over peacefully, which was actually the standard procedure for most cities taken during the Arab Conquests. Our sources sound rather angry, but they were all writers from within the Church and they don't necessarily represent the perspective of the average Christian. Islam was not out to get Christianity and I think it would be reasonable to say that for many people not much changed - people just had to pay taxes to different people (and in many cases the same officials were kept by the new administration, so there was even less of a difference). Ecclesiastical writers tend to view things in a rather apocalyptic way, but once we view things in their context and examine all the facts, it is clear that these authors wildly exaggerate, well, pretty much everything. I'm sure many things did make local Christians angry (Pseudo-Sebeos for example mentioned a Jewish governor of Jerusalem soon after its fall, which must have riled up the local Christians), but it is also clear that mutual accommodation was common as well. Arab soldiers and political leaders had long been active in the Near East and frankly it would be more weird if local Christians were excessively surprised at being bossed around by them.

The reaction beyond the Near East is of course harder to tell and off the top of my head I can't think of a source beyond Fredegar's short description. Jerusalem was certainly important, but to what extent did it matter that it was captured? Pilgrims were still free to go there (we have a very interesting account from Arculf, an Anglo-Saxon pilgrim, for example) and the holy places were still preserved. Plus, Jerusalem had fallen just two decades before and was occupied by the Persians until 628/9, yet there were no doom and gloom predictions about that in the West, so why would anything change now? From what I know of the seventh-century Church, it seems to me that many people in the West were more concerned about the monothelete heresy promoted by Constantinople (to the extent that there were major revolts in North Africa and Italy against the emperor), a dispute that drew in clergymen from Visigothic Spain, Francia and even Anglo-Saxon England. The collapse of Roman authority in the East certainly added a sense of urgency to the actions of radical monks and papal bureaucrats in the West, as they were sure that once their version of orthodoxy had triumphed that the situation would improve, but I can't think of an example of when they explicitly mentioned Jerusalem. For further information, I suggest that you read my comments on the earliest Christian reactions to Islam and on why the Arab Conquests succeeded. Of course, please feel free to ask any further questions as I certainly haven't covered everything and there is definitely space for further debate about this topic!