If European kingdoms united to conquer the Middle East in the medieval time

This army against that army? You must be joking

During then high middle ages, Europe was divided into small feuding kingdoms, duchies and principalities, most as small as 200–300 square kilometres, with the Holy Roman Empire as the largest state (naturally the crusades were largely led by the Holy Romans). Confederated armies of these small states was so ineffective that it took them four centuries conquer back the Iberian Peninsula (the entire Middle East east had 400 times more land and 300 times more people than the peninsula).

By the end of the misadventures of Richard the Lion Heart, Middle East was pretty well organised by Seljuk Empire, Timurid Empire and Abbasid Empire. It would mean a ragtag army without a command structure going against some of the most organised armies in the world.

If you could still gather an army with an effective command structure, it would be clearly have two components - knights (5% of the army) and peasants (95% of the army). None of the peasants went into battle on their own. They did not know how to fight, had no intention to fight, and were very poorly fed and dressed to be effective in a battlefield. That is the reason why in every battle in Europe all belligerent parties lost large number of soldiers very quickly.

Those feuding commanders with the army of unfed peasants, if they could gather, together would then have faced a longest supply line possible over hostile territories. In the way of the army there stood either the Mediterranean or the Byzantine Empire, the biggest adversary of the Holy Romans, up to 14th century, and then by the Muslim Ottoman Empire since 14th century. Getting a horse or a soldier fed, a weapon mended or wounded soldier treated would be perfectly impossible. Every day of fight would take that army closer to annihilation, not victory.

On top of it they had the quick moving, agile Saracen soldiers who lived by the sword from early childhood. If the European army managed to get past the Byzantines and the Seljuk Turks, they would have faced the Mongol and Persian armies of the Timurid Empire, as well as the desert warriors from Bedouin tribes of Arabia, masters of hit-and-run campaigns.

The elite soldiers, i.e. the Teutonic Knights, wore 50 kilograms of metal armour that not only could slow the soldier down by two thirds, and fried the skin of the soldier in sweltering desert heat. The Slavs and Norse were anyways completely ineffective in hotter southern territories. Throughout the medieval times they never won a war in the southern regions. Middle Eastern did not need to fight them, they just could wait for them to boil in the armours or die of hunger and thirst.

The army of peasants would be a very small army compared to the Middle Eastern, as 30%-60% of those hapless peasants were dead in the Black Plague. The entire population Europe was little more than 50 millions, and most of them were children as life expectancy was less than 30 years. They would have been fighting a population of 70–90 millions with a life expectancy of more 35 years. Even if they could gain a quick hold of the coastal areas, a protracted war would only have meant a Europe further depopulated.

Europeans were very good at getting in the medieval times. In the 13th century the Mongol Hordes had already put the entire Eastern European military to shame by cutting them down like grass, knights included. After the Black Plague and Mongol Invasion, these Slavic armies would have been in no shape to join capture one Middle Eastern waterhole, let alone the entire Middle East.

Finally, the people of Middle East would have fought very hard for their homes and their land, as they would have had no place to go to if they lost. Every wadi, every dune, and every village would have become like Stalingrad in WWII, where the might of the Nazi storm troopers stopped at the door of unyielding Russians, a battle that changed the tides of that war.

In short, if the European Kingdoms tried conquering the “entire Middle East” it would mean total destruction of Europe, not the Middle East. Every last European soldier would have either died of hit, hunger, thirst, Saracen swords, and Mongol arrows. With everyone able to lift a sword dead in the deserts of Arabia, Europe would have become an walk over for North African Moors and the Mamulk Sultanate.