It can be observed that the fall of Constantinople to the Turks was generally viewed with terror throughout Europe, as expected; and Pope Nicholas V actually called for an immediate crusade as a response. However, no monarch showed up for it.

What would make a march on the city a bad idea in terms of military strategy and logistics was probably the immense Turkish buffer zone in the Balkans, instead of their actual numbers(which Europe in general likely had no idea of). This is how it looked like before the fall: https://en.wikipedia.org/wiki/Fall_of_Constantinople#mediaviewer/File:Eastern_Mediterranean_1450.svg

Aside from that, Ottomans had already defeated a crusade in Varna very recently, I would imagine that such a defeat would mean little to no support from Poland, Wallachia and Hungary to the monarch that takes up the cause. As "local" Christian military was simply in no shape to assist a massive undertaking.

So Nicholas eventually decided to launch an attack on the city by himself, but died before he could set off. Pope Pius II also declared a Crusade, which shared the same fate of never setting off. After some time, one can presume that the acceptance eventually kicked in.

Hi, I like the question but please understand it is relatively vague and elides a lot of the complexity and nuance in the events of 1453. That being said I will do my best to help clear this up for you a little.

When the Ottoman Empire finally took the city of Constantinople, the "empire" of the Byzantines (or Medieval Romans) was little more than what they could see from the walls of Constantinople. By comparison, the Ottomans had taken much of what is now Greece and Turkey as well as chunks of other territories around the Black Sea and Mediterranean; the Byzantines were unable to call up any more people and could ill afford to pay for a mercenary army. Part of this is because for quite some time (many decades in fact) the Byzantines had been paying the Ottomans protection money in the form of tribute.

Add to that the fact that since the 4th crusade when the "Latins" (that is Latin Christians) sacked Constantinople instead of attacking the Turks, that the Latins and the Byzantines did not like each other very much. Even though the Byzantine Emperor did call for help against the Ottoman Turks, there was very little empathy for the "heretics" of Constantinople.

Even if the Latins did want to help, they would not have been able to fend off the Ottomans for very long; the Ottomans were considered a 'gunpowder empire' with good reason; they dove head-first into gunpowder adoption AND had the money to field a fairly large army armed with the latest in weaponry, including a cannon that fired 1/4 ton boulders. It was so large, in fact, that evidence suggests that Italian city-state merchants like those from Genoa were "coerced" into helping the Ottomans, even though the Italians had relatively close economic ties to the Byzantines. Basically, the Europeans who were near enough saw the "writing on the wall" and opted to stay out of the fight.

The Vatican wasn't interested in directly fighting the Ottomans; it had its own problems too: barely a generation before, the Western Schism split the papacy and its political power, including the ability to call for crusades. That is not to say, however, that the papacy didn't fuel local resistance to Ottoman encroachment into Europe - they did, including funding Vlad Tepes, also known ad Vlad Dracul, in his war against the Ottomans. The Pope also funded others, including Hungarians, but the money didn't exactly go towards fighting the Ottomans and may have strongly contributed to the loss the Christians suffered in the Battle of Varna: basically, the Ottomans were too well disciplined and too well armed to be defeated by the combined Polish-Hungarian army that could be mustered at the time.

As for the rest of Europe; they had bigger fish to fry. Namely, each other. The French and English had exhausted each other with the Hundred Years' War; the Spanish and Portuguese were busy wrapping up the Reconquista, and the Holy Roman Empire had recently put down the Bohemian Rebellion (also called the Hussite Wars), so were also not in a position to aid a "heretic" like the Emperor Constantine XI.

As for how the Ottomans acquired such a big army, the answer is two-fold; first, a lot of the money that was paid by the Byzantines to the Ottomans went to swelling the ranks of the army. This included the forging of top-of-the-line gunpowder weapons and cannons, as well as more mundane things like procuring more mercenaries (as well as Christian boys who became Janissaries). Second, the Ottomans' culture was a warrior culture, what the Turks call ghazi, in which warfare, especially against the infidel Christian and traditional enemy of Islam, the Byzantines, was considered an honor. Many flocked to become a member of the army not only to establish themselves as a ghazi but also to take some of the loot which was part-and-parcel of the pillaging of the enemy's lands. Money and glory, basically.

The wider "European" world probably didn't realize or care about the threat from the Ottomans until the first siege of Vienna in 1529, but by then, it was too late; the Ottomans had conquered so much territory, they seemed invincible and only through a combination of tenacity of the Austrians and the mutinous attitude of the Janissaries kept Vienna from falling. Though the battle ended with the Ottomans not taking Vienna, they did exact tribute. In addition, by that time, the Ottoman Empire was far, far more powerful than any one (or combination) of countries' armies. Even the "Battle of Lepanto" - a naval battle in which the Catholic navy of a coalition of European countries smashed the outdated Ottoman fleet - was only a temporary setback for this vast, rich empire. As Mehmet Sokullu noted to the Venetian Ambassador a year or so later that the defeat was inconsequential "You come to see how we bear our misfortune. But I would have you know the difference between your loss and ours. In wresting Cyprus from you, we deprived you of an arm; in defeating our fleet, you shaved our beard. And arm when cut off cannot grow back; but a shorn beard will grow all the better for the blade." Indeed, he was right in that within 6 months after the disastrous encounter with the Christian Catholic coalition, the Ottoman navy was bigger, better, and more modern than it had ever been before and the Ottomans would hold onto Cyprus for centuries.

All in all, the upshot is that Western and Central Europeans were not only busy with their own wars, but they lacked the wherewithal and inclination to do anything but watch the sick old man that was the Byzantine Empire be put out of its misery. By the time they realized their mistake, it was too late and the Ottomans were too rich, too big, and too powerful to defeat. For now.

Ottomans were using a land-granting system named "Tımar", which transfers the burden of the raising an army to local rulers called "Bey"s. In Tımar, small landowners named Timariots were obliged to raise soldiers and join their Bey's armies themselves in return of a piece of land granted to them. When you consider that a significant percentage of the Ottoman population were actually land-workers at that time, you can guess how big and easily mobilized that army was.

After Istanbul's conquest, Tımar system started to decline and it slowly transformed into a "land renting system" governed by the State.

Sources:

Halil İnalcık - “Ottoman Methods of conquest.”

Donald Quataert and Halil İnalcık - "An Economic and Social history of the Ottoman Empire"

By the time of the siege of Constantinople the Ottoman Empire had already taken much of the Balkans and their capital was based in Edirne from 1365 CE to 1453 CE. The Ottomans were already a force to be reckoned with. Under Murad II, the Ottomans beat a Polish-Hungarian coalition and forced a peace treaty on them that added the Kingdoms of Bosnia and Wallachia to the Ottoman sphere of influence. In 1453, Constantinople fell after Mehmed II defeated the Byzantine army under the command of Constantine XI. The Ottoman army probably numbered over 100,000 and some accounts state it was as large as 300,000, but these accounts are widely considered exaggerations. Within the century, the Ottomans had taken the last vestiges of Byzantine Rule by taking the rest of Greece.

There is a problem with your question though. It should not be how were the Ottomans able to raise such a large army, but rather why did European Christians not raise an army to fight the Ottomans? The simple answer is there was a lack of desire to fight the Ottomans. Europe knew the Ottomans were powerful and were solidified in Anatolia and the Balkans, meaning any attack would be prone to disaster. Pope Nicholas V asked the Christian rulers of Europe to step up and retake Constantinople, but no one would take on the mantle. At the time Europe was fragmented and its kingdoms were too involved with infighting to have the strength to launch an overseas crusade. England was in the middle of the War of the Roses, the Rhineland was involved in a war in Liege, the rest of the HRE was involved in regional conflicts, Austria and Hungary were in the middle of tensions that would eventually lead to a war, and France was still trying to solidify its rule. In lieu of this, the pope decided to lead the campaign himself, but his untimely death removed the last leading figure in Europe that was willing to go to Istanbul and retake it.

I will provide sources later since my laptop is about to die so it will be a bit.

EDIT: Also Iberia was still in its reconquista phase.

I'm not sure I can answer in the specific context of the size of their army, but when Venetian and Genoese armies were present at the battle for Constantinople. After the fall though, no European power had any vested interest in fighting a far off land. The crusading fervor of previous centuries had mostly passed, as Europe was largely focused on itself. France and England were fighting the Hundred Years' War, the HRE was mostly too decentralized to field any large army, and Poland and Hungary were still wary of engaging the Ottomans, who had beat them at Varna previously. As time passed, the Ottomans stopped posing of large a threat, and Europe didn't care too much about them anymore. That is until the end of WWI, when France and Britain placed Istanbul under their control, as they didn't want it's strategic location to end up in the USSR's sphere. It could be said that WWI was when Europeans finally retook Constantinople.