The letter, organized by South Carolina freshman Jeff Duncan, said Rice's "misleading statements" about the attack that led to the deaths of Ambassador Chris Stevens and three other Americans "caused irreparable damage to her credibility both at home and around the world."
It was the latest GOP effort to single out Rice for the mixed signals sent out by the administration in the immediate aftermath of the September attack in Benghazi.
Republican Sens. John McCain and Lindsey Graham have led criticism in the Senate, saying Rice is unqualified and untrustworthy and promising to block her nomination if Obama picks her to take over the State Department after Clinton steps down.
Obama responded last week at a news conference, saying McCain and Graham should "go after me" if they want to criticize administration actions. He said Rice had nothing to do with the Benghazi affair and "to besmirch her reputation is outrageous."
On Friday a dozen Democratic female members of the House also came to Rice's defense, saying the criticisms of her smacked of sexism and racism.
Rice became a target when she went on the Sunday talk shows five days after the attack on the Benghazi diplomatic mission on Sept. 11 and said that, from the best information she had at the time, the attack was a spontaneous reaction to an anti-Muslim video and not a premeditated attack. That assessment later proved to be incorrect.
Rice, the House Republicans said in their letter, "is widely viewed as having either willfully or incompetently misled the American public in the Benghazi matter." As a result, they said, "we believe that making her the face of U.S. foreign policy in your second term would greatly undermine your desire to improve U.S. relations with the world and continue to build trust with the American people."
Clinton has not formally announced when she is leaving her post and Obama has not said who might succeed her, although Rice has often been named as a leading candidate. Senior-level positions must be confirmed by the Senate.