After Star Trek III: The Search of Spock, the resurrected Vulcan was supposed to have a baby with Lt. Saavik (Robin Curtis), but the entire storyline was deleted from canon. The third Star Trek movie was directed by Spock himself, Leonard Nimoy, and involved Admiral James T. Kirk (William Shatner) and the crew of the Starship Enterprise disobeying orders to retrieve the late Vulcan's body from the Genesis Planet. But the major subplot about Spock and Saavik conceiving a child together in The Search for Spock was dropped during the making of the next film, Star Trek IV: The Voyage Home.
Star Trek II: The Wrath of Khan was a critical and financial success that culminated with Spock heroically sacrificing his life to stop Khan (Ricardo Montalban) from destroying the Enterprise with the planet terraforming Genesis Device. After his touching funeral, Spock's body was placed in a photon torpedo and launched towards the newly-forming Genesis Planet, where it landed safely unbeknownst to the Enterprise's crew. In Star Trek III, the U.S.S. Grissom was sent to survey the Genesis Planet; Saavik and Kirk's son David Marcus (Merritt Butrick), who helped build the Genesis Device, beamed down to the surface and discovered that Spock wasn't just resurrected but the Genesis effect de-aged him and he was growing rapidly from an infant into adulthood with no knowledge of who or what he was. When Spock aged into adolescence, he underwent Pon Farr, the painful Vulcan mating ritual. Understanding what this meant, Saavik has sexual intercourse with the young Spock to alleviate his suffering — and originally, this resulted in Spock getting Saavik pregnant.
In the early plans for Star Trek IV, the reason Saavik remained on Vulcan at the start of the film instead of rejoining Kirk's crew on their mission to save the Earth was that she was carrying Spock's child. Additionally, the producers felt writing Saavik into Star Trek IV's main story was too difficult, given how the entire Enterprise crew was going to time-travel back to 1986 in that crowd-pleasing adventure. In the audio commentary of the Star Trek IV: Directors Cut DVD, director Leonard Nimoy explained that it seemed "more interesting to leave her behind with the potential information that she was expecting Spock's child." However, by the time Star Trek IV was completed, Saavik's pregnancy was cut out of the movie — and thus, eliminated from Star Trek canon entirely. After her brief appearance in Star Trek IV, Saavik was not canonically seen again.
Spock and Saavik having a child together was deleted because, in producer Harve Bennett's words, Nimoy "was always very uncomfortable" about it. As the director who also happens to be Spock, Nimoy had the clout to make the final decision to cancel Spock being a father. According to Bennett, scenes about Saavik's pregnancy from various stages of the screenplay were excised from Star Trek IV. In the initial screenplay, Saavik takes maternity leave from Starfleet on Vulcan without telling Spock about her condition. In the final version of Star Trek IV, Kirk, Spock, and their crew depart Vulcan for Earth, simply leaving Saavik behind on Spock's homeworld. But since her pregnancy isn't acknowledged, it always seemed strange that Saavik, who is also a Starfleet Officer, elected to remain on Vulcan.
Had Saavik's pregnancy storyline followed through, it's unclear what the plans were, if any, for Spock finding out he was a father and what would have happened next. Ultimately, however, the point is moot since the whole affair was dropped. Indeed, it would have been fascinating to explore how Spock would have fared as a father, but the franchise never went in that direction. Since Star Trek III killed off Kirk's son David, it meant that Hikaru Sulu (George Takei) is the only member of The Original Series' Enterprise crew who canonically became a parent; Sulu's daughter Demora (Jacqueline Kim) was introduced as the helmsman of the U.S.S. Enterprise-B in Star Trek Generations. Sadly, Spock died without ever siring an heir, and in Star Trek canon, neither Kirk nor Spock got to have descendants to carry on their adventures and save the galaxy.