The public clash also comes at an awkward time for Fox News, which will celebrate its 20th anniversary on Friday and has been trying to move forward from the ouster of its longtime chief, Roger Ailes, after several claims of sexual harassment.
Kelly did not immediately respond to Mr.
Hannity, a network stalwart, is close to Mr.
Kelly, a newer addition, became one of Fox’s biggest stars by appealing to a younger audience, and she is widely seen as an important figure for the network’s future.
Asian-American groups denounced the segment as racist, citing its copious references to Asian stereotypes.