Short version: no. I'm actually really curious where you heard this, because I've never encountered this story. I've heard the 'Richard I was gay' thing before, which has problems all on its own, but never anything like this.

Longer version: There's a whole side discussion to be had about how to define failure with regard to the Third Crusade but for the moment I'll skip past that and assume that we're okay with the idea that since it failed to retake Jerusalem it was, at least on a grand scale, a failure.

The failure to retake Jerusalem, however, cannot be placed entirely on the feet of King Philip's early departure from the crusade. Historical consensus generally seems to be that Philip was never particularly invested in Crusading and saw time away from his work in France (which, among other things, consisted of undermining Angevin power and bolstering his own authority) as an unnecessary distraction. Whether his participation in the Crusade can be seen as his piety overwhelming his better judgment, but only briefly, or as a tactic to simply appease the Pope while actively engaging in conflicts with other European kings can be debated forever. The short version is he showed up for the Siege of Acre, and left pretty much as soon as it was over.

In some ways Philip's departure was beneficial to the Third Crusade. While the loss in manpower was obviously detrimental, it left Richard as the clear leader. While King Guy could in theory claim some leadership role, his disastrous performance at Hattin made him only a moderate problem. This is not to say that everything went smoothly, the problems with Conrad of Montferrat and the rumors that Richard had him killed are sufficient evidence of that, but the presence of Philip II would certainly not have made things smoother from a leadership perspective. Richard's longer time in the Holy Land was actually reasonably successful, his victory at the Battle of Arsuf and retaking of the port of Jaffa were significant, but he was never quite able to challenge Saladin's defensive position in Jerusalem. Richard's greatest problem in this regard was that remaining in the Holy Land with his army was draining his royal coffers, while Saladin had a local financial base and could simply live in Jerusalem and wait for Richard's funds to run out. While initially the promise of vast reinforcements via the arrival of Frederick Barbarossa promised the possibility of an assault on Jerusalem, the death of Barbarossa en route and the dismantling of his army doomed that campaign far more than Philip II's departure ever did.

While the exact reason for the Third Crusade's failure to retake Jerusalem can be debated ad nauseum, the sole blame of it cannot be loaded onto Philip's early departure, and that departure was not the result of any sort of awkward semi-romantic moment between him and Richard.

Thomas Asbridge's The Crusades is a great source for this period. Anne-Marie Edde's Saladin also offers some great information on the Third Crusade but from Saladin's perspective, obviously.