It's not entirely wrong, but I wouldn't agree with the framing of it. There's a big caveat here that we can't truly know what Saladin thought or believed. He left very few written documents in his own hands - and none of them are introspective diaries or anything that provides a deep insight into his inner thoughts. Instead we have to construct his opinions by biographies written by individuals who were generally either very hostile to him or falling over themselves to praise him.

That said, it does seem that Saladin was frustrated with his fellow Muslims around the time of the Third Crusade. Some context is important. Despite Saladin's reputation as a foe of Christianity and the Crusader States he spent far more of his life fighting Muslims instead of Christians. Upon the death of his uncle Shirkuh Saladin had taken control of Egypt - at the time the richest province in the region - and looked like he was about to rebel against his lord Nur al-Din. There's some debate as to whether this was a genuine rebellion or if it was a case of a misunderstanding going to far, but it all became a bit moot because just as he was summoning an army to crush Saladin Nur al-Din died, leaving a teenage son to inherit. Saladin then began a war of conquest to take over all the territory that Nur al-Din had ruled - ostensibly arguing that he was doing so to preserve the lands of Nur al-Din's son and to save him from Nur al-Dins' "treacherous advisors" who were acting as regents. This is pretty standard dynastic disputes and conflict. Saladin was reasonably successful in his conquests and when Nur al-Din's son died some years later Saladin was well positioned to take over all of the Syrian lands that had once belonged to his master. It was only after this point (and some conquests in Yemen and North Africa as far west as modern day Tunisia) that Saladin turned his focus to the Crusader States and Jerusalem (it's worth noting that Saladin had clashed with Christians periodically during this earlier period, it was just never his primary focus and had little lasting impact on either side).

When Saladin took Jerusalem and the Third Crusade began to assemble to retake it he was in a tough spot. He'd been campaigning near constantly for decades and his resources and soldiers were at their limit. Realistically he didn't have much of a chance of opposing a united Christian army - in open battle anyway. He reached out to his Muslim neighbours in search of support, most notably the Seljuk Turks in Anatolia (modern day Turkey) and the Abbasid Caliph in Baghdad. The Caliph offered some token support, but nothing much of substance ever materialised, while the Seljuks pretty much ignored Saladin entirely. There's a lot to unpack to fully explain this response, but the short version is that they saw Saladin as potentially a greater threat to them than the Crusaders and weren't particularly worried about seeing him taken down a few notches. Neither faction was so opposed to him that they would consider supporting the Crusader armies, but they were happy to not interfere if they could get away with it (the Caliph was far enough away geographically to be safe, the Seljuks were largely spared involvement since the Third Crusade arrived primarily by ship rather than marching through Seljuk territory from Byzantium).

In the end, the death of Frederick Barbarossa en route to Crusade probably spared Saladin some of his greatest risk (Frederick's army was reported to be enormous) and also spared the Seljuks from having to decide whether to get involved or not (Frederick would have had to march through Seljuk held territory to reach the Holy Land). It's also worth noting that Christian unity wasn't all it was cracked up to be, Richard I and Philip II did not get along and Philip II didn't linger in the Holy Land but rushed back to France to continue his campaigns to undermine the English king's rule over French territories. There's still endless room to debate if Richard I could have beaten Saladin had he tried (as I mentioned Saladin was in a rough spot in terms of finance and man power) but in the end he never attacked Jerusalem so we won't know.

I would agree that Saladin was frustrated with some of his fellow Muslims over their failure to support him, but I don't really agree with the framing that he was envious of Christanity - I think that makes it seem like he was petulant or felt himself inferior to them (but maybe I'm reading too much into this). I would also add that at this time Saladin ruled over a huge proportion of the Muslim world - so it's not really fair to say that he lacked the support of the Islamic World, in many ways he practically was the Islamic World (at least in Western Asia and North Africa), it was instead his political rivals who didn't support him. Saladin's own territories were very supportive of his wars.

Anne Marie Edde's biography Saladin is excellent and goes into lots of detail on Saladin's relationships with other Muslim rulers and Christianity in general. It's probably not a great entry level book, though, and I'd recommend reading something more general like Thomas Asbridge's The Crusades before tackling it.