The Fresno Bee preserved a screen grab of Jarrar's tweets and wrote about the ensuing back-and-forth with outraged critics before Jarrar made her account private.
Jarrar also reportedly tweeted, "I'm happy the witch is dead. can't wait for the rest of her family to fall to their demise the way 1.5 million iraqis have. byyyeeeeeee."
In his statement, Castro wrote that while the professor's conduct "was insensitive, inappropriate and an embarrassment to the university," Jarrar was nonetheless "acting in a private capacity and speaking about a public matter on her personal Twitter account. Her comments, although disgraceful, are protected free speech under the First Amendment of the U.S. Constitution."
Jarrar had welcomed critics to reach out to her employer, including the Twitter handles of Fresno State and its president in a tweet that read in part, "What I love about being an American professor is my right to free speech, and what I love about Fresno State is that I always feel protected and at home here."
Her tweetstorm generated thousands of responses, the Bee reports, many of them demanding that Fresno State fire Jarrar. As if to underscore that this would never happen, Jarrar noted that she is a tenured professor in the English Department and makes $100,000 a year, according to the newspaper.
But Castro said the decision to allow Jarrar to keep her job wasn't related to tenure, saying it "is an issue of free speech."
He went on to say, "Let me be clear, on campus and whenever we are representing the university, I expect all of us to engage in respectful dialogue."
Lars Maischak, an untenured history lecturer at Fresno State, bumped up against those limits last year when he tweeted that "Trump must hang," and then made a reference to a "Trump assassin."
Following a backlash, Maischak was put on leave and Fresno State announced he would not be teaching on campus in the fall of 2017.
Maischak now teaches courses online.
Following Castro's announcement, the American Civil Liberties Union, which had come to Jarrar's defense, tweeted, "Randa Jarrar's tweets about Barbara Bush are protected speech under the First Amendment. Period."
But some people posting under the hashtag #boycottfresnostate said Jarrar's comments went too far and lamented the lack of disciplinary action taken against her.
A petition on change.org calling for Jarrar to be fired had garnered more than 60,000 signatures as of Wednesday afternoon.
For her part, Jarrar, who is also an author, has been keeping a lower profile.