Muslim armies defeat the Mongols in the Siege of Baghdad

As it is known, the Mongol's attempted conquest of most of Eastern Europe was thrwarted by the death of Genghis Khan (at least, that was the excuse) and the troops had to go back to Mongolia to reorganize. However, the Mongols were able to defeat the Abbassid Caliphate in 1258 at Baghdad, causing the downfall of the already declining Islamic Caliphate. What would have happened if the Mongols had been decisively defeated at the Siege of Baghdad? This could possibly coincide with the death of Hulegu Khan of the Ilkhanids, throwing the Mongol armies off.
Could the pan-Arab Islamic Caliphate be revived by the new morale that accompanies an Islamic/Arab victory at Baghdad? Would the Islamic Golden Age last? Would the Islamic world have a similar history to that of post-Medieval Europe (renaissance, expansion)?
 
As it is known, the Mongol's attempted conquest of most of Eastern Europe was thrwarted by the death of Genghis Khan (at least, that was the excuse) and the troops had to go back to Mongolia to reorganize. However, the Mongols were able to defeat the Abbassid Caliphate in 1258 at Baghdad, causing the downfall of the already declining Islamic Caliphate. What would have happened if the Mongols had been decisively defeated at the Siege of Baghdad? This could possibly coincide with the death of Hulegu Khan of the Ilkhanids, throwing the Mongol armies off.
Could the pan-Arab Islamic Caliphate be revived by the new morale that accompanies an Islamic/Arab victory at Baghdad? Would the Islamic Golden Age last? Would the Islamic world have a similar history to that of post-Medieval Europe (renaissance, expansion)?

You're getting confused here. It was the death of Ögedei, Genghis' son, in 1241 that led to the Mongol withdrawal from Europe, and this didn't really do much to seriously damage Mongol military potency as you seem to be implying.

By 1258, there hadn't been an "Islamic Caliphate" in the sense of a single unified Islamic state for centuries; indeed, the Arabs themselves had been conquered by the Turks when the Seljuks swept all before them in the tenth and eleventh centuries. The power of Turkish and Iranian rulers by the thirteenth century compared to the eighth is going to make ideas of a revived pan-Islamic world problematic, to say the least.

The Islamic world is unlikely to enjoy the fruits of the reformation and enlightenment, I would say, regardless of whether Baghdad is sacked or not. Like the Roman Empire before it, the Arab Caliphate was basically a savage despotism, despite its high culture and learning.
 
Indeed, the Middle East had been divided up by Turkish Slave Soldiers turned Lords for a few hundred years by this point.
 
A better POD would be the Kwarazmians doing better and holding off Mongols long enough to sap their strength. After Chingiz's death, the sons would quarrel over the much lesser patrimony.

Other than that I can't think of a coherent military Muslim power that can put up much resistance unless seriously aided by geography (see India, Egypt).
 
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