"History will judge them as having stood by a brutal dictator at the expense of his own people and at the expense of the will of the international community and the countries in the region," Rice told CNN.
The fate of the U.N. observer mission was unclear after the vetoes.
Competing drafts, sponsored by the United Kingdom and Pakistan, are expected to be considered by the U.N. Security Council. They highlight the simmering differences among members.
The UK mission said its resolution will be considered Friday morning. That draft measure extends the observer mission in Syria for a final period of 30 days.
Russia, which backs Pakistan's resolution, wants observers to stay in the country and promote dialogue. Western countries are concerned about the safety of observers and want one final deployment.
China's U.N. ambassador, Li Baodong, said his country is "highly concerned" about the violence, but Thursday's draft resolution was "seriously problematic," because it intended to pressure only one side in the conflict. It would not help resolve the issue "but derail the issue from the track of political segment," further aggravating turmoil, he argued.
Syria's U.N. ambassador, Bashar Jaafari, slammed the council for not issuing an official condemnation of Wednesday's bombing, although he noted that Kofi Annan, joint envoy to Syria for the United Nations and the Arab League, condemned it.
The lack of an official condemnation by the council indicates that talk of "supporting a peaceful solution in Syria is but a slogan," Jaafari argued.
He rejected a "misleading picture" of Syria as a tyrannical regime killing its people and said a solution should be found through a "political process that satisfies the Syrian people." All will participate to help "establish democracy" and "free elections," he insisted.
The al-Assad family has ruled Syria for more than 40 years.
Residents trying to flee the Syrian violence have nowhere to go because clashes are raging in most of Damascus province, said Omar al-Dimashki, a spokesman for the Revolution Leadership Council of Damascus.
"It is surrounded by tanks, and anything that moves is currently being shelled, and rockets are falling on the homes," he said.
Sander van Hoorn, a journalist in Damascus with Dutch TV network NOS, a CNN affiliate, said, "One hour it can be quiet and the next hour all hell breaks loose."
A resident of the al-Qaboun neighborhood in Damascus said he fled with his wife and two sons after an explosion shook a house next door.
The man, who CNN is not identifying for security reasons, said his sister and her family joined them in Qamishli, and she described the government assault on al-Qaboun. "There were tanks, helicopters, armored personnel carriers and gunshots fell like rain. We were terrified," he quoted her as saying.
The Free Syrian Army warned them the neighborhood was "going to be a war zone."
Syria, which blames the violence on "armed terrorist groups," said it "repelled" some groups that attempted to enter Syria from Lebanon on Wednesday night.
The regime said Wednesday's bombing was carried out by people "implementing foreign plots."
The Syrian military issued a statement stressing its "resolution to decisively eliminating the criminal and murder gangs and chasing them out of their rotten hideouts wherever they are until clearing the homeland of their evils," state-run news agency SANA said.
Anyone who thinks that targeting leaders will "twist Syria's arms is 'deluded,'" the statement said, according to SANA.
The officials killed in the blast were the defense minister; the deputy defense minister, who is the president's brother-in-law; and al-Assad's security adviser and assistant vice president, state TV reported.
The FSA's al-Kurdi said the attack was coordinated by rebel brigades. But some other rebel commanders say it's unclear who was behind the attack.
Since the crisis began in March 2011, the United Nations estimates, more than 10,000 people have been killed in the violence; the opposition Local Coordination Committees of Syria estimates that more than 16,000 have died.
CNN cannot independently confirm reports of violence in the nation because the government restricts access by foreign journalists.