A very elaborate ending was indeed scripted and filmed for the theatrical version of The Abyss. It starts off after Bud is saved by the aliens. They show him the news on a giant video wall about the imminent war between the USA and the Soviet Union. This changes into a special report about giant tidal waves that appear everywhere around the globe, threatening all mankind. Bud instantly understands that the aliens are responsible for these tsunamis, and he asks why they want to destroy mankind. They respond by showing him images of the many wars and atrocities committed by humans in their seemingly never-ending route to self-destruction. But, as the waves are about to hit the land, they stop and retreat; the aliens have stopped them. Bud's sacrifice has showed them that humanity may have the strength and wisdom to make it after all. The movie then ends with the spaceship rising up towards the ocean surface.
This ending, together with many other deleted scenes, was restored in the Special Edition of The Abyss. The reason that these scenes were deleted from the earlier version is that they would have made the movie almost 3 hours long. In those days, a running time that long was generally considered too big a commercial risk, especially for big-budget, action-packed, special effects-heavy blockbusters that have to earn a lot of money to break even, let alone make a profit. A general rule is that movies make the most money in the first week(s) or weekend after the premiere. A long running time has the disadvantage that cinemas can hold fewer screenings per day, which diminishes the box-office per day of showing. Furthermore, conventional wisdom at the time was that audiences were not very enthusiastic about remaining focused for such a long time, following the commercial failures of Novecento, Heaven's Gate and Once Upon a Time in America; a notion that was largely disproved since then, with many fine examples of 3-hour-plus movies that became huge box-office hits, e.g. Dances With Wolves, Lord of the Rings, James Cameron's own Titanic and Avatar. Also, it was originally released during the summer film season of 1989, albeit very late in the season, and most summer films then didn't run much longer than 2 hours.
Contrary to what may be thought, the omission of the wave scene was not a studio decision; director James Cameron and producer Gale Anne Hurd had final cut as long as the movie was kept under an agreed-upon length of 135 minutes, and according to Cameron, the studio was even disappointed to see the wave scene go. However, the special effects of the scene had not been finished, and reactions from test audiences to the unfinished scene were mixed. So Cameron and Hurd were faced with the decision to either trim shots and parts of scenes away from other parts of the movie, or remove an entire subplot. They decided upon the latter, feeling that the first option would hurt the movie's focus and character development, and they also thought that the movie's theme still work without the wave scene, although less overtly. So the tidal wave scenes were never completely finished during post-production.
However, a favourable response to the Special Edition of Cameron's Aliens in 1991 prompted him to revisit The Abyss in 1993 and restore the deleted scenes with fully-finished special effects and a slightly re-recorded soundtrack.