Rich, 33, and a SNL writer since 2013, deleted her Twitter account after the pushback but revived it Monday afternoon to post an apology.
Clinton - who was just 13 when her father became President in 1993 - took to Twitter to defend Barron Trump after he became the butt of jokes on social media during his father's inauguration.
"Barron Trump deserves the chance every child does - to be a kid", Clinton wrote.
Rich's name did not appear in the credits on the most recent original broadcast of "Saturday Night Live", which aired on NBC Saturday.
Barron Trump, walking in the inaugural parade.
"That it even need be said is ridiculous, but mocking the 10-year-old child of a politician you loathe is odious, immoral, & self-defeating", tweeted CNN's Jake Tapper.
People immediately took such jokes to task, considering they involved a kid who didn't ask to be in the spotlight, and children in the first family are supposed to be off-limits.
Miss Clinton drew on the eight years of her childhood in the White House when she Tweeted about how hard it could be to be the child of a serving President.
Some criticized the post for quickly attacking potential policies Trump could enact as POTUS, but others have since remarked positively on her comment because she was encouraging people to lay off Barron.
Barron Trump, 10, has a defender in former First Daughter Chelsea Clinton, 36. Nearly immediately, Rich began getting a slew of angry tweets in response to her comment.
Despite a few critics, most social media users supported Chelsea's message.
The Friday tweet prompted a rash of online petitions calling for NBCUniversal to fire Ms. It is presumed that Melania and Barron Trump will move into the White House sometime over the summer. She posted: "I think Barron is on his Gameboy".
In the 2014 interview with Thomas, Clinton said that through it all, her parents wanted her to maintain a normal life, which is what she hopes for Barron Trump.
President George W. Bush's daughters, Barbara and Jenna, wrote a letter to President Barack Obama's daughters, Sasha and Malia, when they entered the White House.
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