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Thing is, the concept of the beach remaining a secret at any and all cost had reached the point where it completely warped everyone's sense of judgement, Richard's included (it's why Sal was so confident at the end). To him, telling Francoise the reason he had sex with Sal (so she would keep quiet about him copying the map) was by that point worse than Francoise finding out he had cheated in the first place; notice how when Sal leaves him on the hill, he's mostly just concerned about her not telling anyone that he copied the map. There's also the fact that Richard and Francoise's relationship was kept incredibly shallow - they were both young and attractive...and that's pretty much it. The scene with Keaty telling Richard the 7 points of incompatibility with Francoise, while primarily a joke, had quite a lot of truth in it. There was no real basis for a "true love" long-term relationship: Richard was too emotionally immature and even when she was with Richard, Francoise clearly still cared a lot about Etienne, who incidentally was ultimately shown to be the better man in the entire scenario than Richard (he's pretty much the only one in the whole movie who never screws anyone else over and is fundamentally a good person).

Richard's relationship with Francoise was basically a parallel to their experience on the island: fun while it lasted but ultimately unsustainable without getting other people (and themselves) hurt along the way.

Why should he? After all, Francois had cheated on Etienne with him. So what goes around comes around.