Friendship 7's splashdown was mundane.
The Right Stuff depicted two splashdowns. First they showed Shepherd's Freedom 7 splashdown, because it was the first one and thus held some suspense. The scene showed the audience what a splashdown looks like and what happens when everything goes right. Second, they showed Grissom's Liberty Bell 7 splashdown. This time, there was a mishap at splashdown and therefore that too was a suspenseful scene, worthy of the audience's time.
By the time we get to Friendship 7, the audience has seen a good splashdown and a bad splashdown. Since the splashdown of Friendship 7 was pretty much uneventful, showing it would have been a waste of time from a story-telling (and run-time) standpoint. The audience had already seen a good splashdown with the first one.
The drama of Friendship 7 was in the many orbits, the warning light ground control received, and the rocky re-entry, all of which were depicted in detail in the film precisely because they were dramatic.