Fox TV instead showed repeats of the shows to avoid broadcasting any potentially sensitive content.
The billed Family Guy episode had featured a retelling of the nativity while in American Dad, a demon punished naughty children at Christmas.
Meanwhile, director Quentin Tarantino has defended violence in movies.
Tarantino’s latest film, spaghetti western Django Unchained, features graphic violence, including buckets of blood exploding from characters as they are shot.
He said at a press junket in New York for the film on Saturday that he was tired of defending his films each time the US is shocked by gun violence.
“I just think you know there’s violence in the world, tragedies happen, blame the playmakers,” he said, adding: “It’s a western. Give me a break.”
Django Unchained is nominated for five Golden Globes at next year’s awards.
The Oscar-nominated director, whose work includes Inglourious Basterds and the Palme d’Or winning Pulp Fiction, said blame for violence should fall on those guilty of the crimes.
But lead actor, Jamie Foxx, said he believes big-screen violence can influence people.
Foxx said: “We cannot turn our back and say that violence in films or anything that we do doesn’t have a sort of influence. It does.”
Foxx stars as the protagonist of Django Unchained, a slave living in the Deep South who sets out to rescue his wife from a brutal Mississippi plantation owner, played by Leonardo DiCaprio.
Actor Christoph Waltz, who is also in the film, said he thought films did not provoke violence.
“The media’s responsibility is greater than the story teller is because… Django is violent, but it’s not inspiring violence.
“Because actually to me I find violence… to that degree repulsive. The fact that it looks so impressive is because it’s on a big screen.”
He added that violence was part of the film because it tells the story of American history.
Kerry Washington, who plays Django’s wife Broomhilda, said she believes the film’s explicit brutality serves an important purpose in educating audiences about the atrocities of slavery.
“I do think that it’s important when we have the opportunity to talk about violence and not just kind of have it as entertainment, but connect it to the wrongs, the injustices, the social ills,” she said.
On Saturday, premieres in Pittsburgh and Los Angeles for Tom Cruise’s new action film Reacher and Billy Crystal comedy Parental Guidance were postponed in light of the shootings.
Paramount Pictures said the decision for Jack Reacher was made “out of honour and respect for the families of the victims whose lives were senselessly taken”. The film opens with a sniper shooting several people dead.
Fox said Parental Guidance was postponed “in light of the horrific tragedy”.
“The hearts of all involved with this film go out to the victims, their families, their community, and our entire nation in mourning.”
Fox also confirmed that a scheduled repeat of The Cleveland Show for Sunday was swapped for a repeat to avoid any potential insensitivities.
Twenty children and six women died in the assault on Sandy Hook school by a lone man who then took his own life.
The gunman has been identified by police as Adam Lanza, 20. He shot dead his mother before driving to the school in her car.