Speaking in Bulgaria, Mrs. Clinton said efforts outside the world body to help Syria's people should be redoubled.
The US, she said, would work with "friends of a democratic Syria" to support opponents of Syria's president.
The vetoing of the resolution drew an angry reaction from around the world.
"What happened yesterday at the United Nations was a travesty," Mrs. Clinton said in strongly worded remarks during a visit to the Bulgarian capital, Sofia.
"Faced with a neutered Security Council, we have to redouble our efforts outside of the United Nations with those allies and partners who support the Syrian people's right to have a better future," she said.
Analysts say Mrs. Clinton appeared to be alluding to the formation of a grouping of nations similar to the Contact Group on Libya. That group - a collection of Arab and other countries - oversaw international help for opponents of former Libyan leader Muammar Gaddafi.
Mrs. Clinton also said the US would work to tighten "regional and national" sanctions against Syria to hamper its ability to use arms against the protesters.
"We will work to expose those who are still funding the regime and sending it weapons to be used against defenseless Syrians, including women and children."
French Foreign Minister Alain Juppe called the Russian-Chinese veto a "moral stain" on the UN. He said Europe would strengthen sanctions against Syria and eventually "the regime will have to realize that it is completely isolated and cannot continue".
At least 28 civilians were killed by security forces across Syria on Sunday, mainly in the central city of Homs, said the London-based campaign group Syrian Observatory for Human Rights. One report quoting the group said an equivalent number of Syrian troops were also killed.
Human rights groups and activists say more than 7,000 people have been killed by Syrian security forces since the uprising began in March.
The UN stopped estimating the death toll in Syria after it passed 5,400 in January, saying it was too difficult to confirm.
The government says at least 2,000 members of the security forces have been killed fighting "armed gangs and terrorists".
'License to kill'
The draft resolution - which had already been watered down in an apparent attempt to overcome Russian objections - was supported by 13 of the 15 members of the UN Security Council, when it was put to a vote on Saturday.
It was the second time in four months that Russia had obstructed a resolution condemning Syria.
Russia's Deputy Foreign Minister Gennady Gatilov said the compromise over the text did not go far enough, saying its authors "did not want to undertake an extra effort and come to a consensus".
China's state news agency Xinhua said the two countries believed more time and patience was needed to solve the crisis in Syria.
The move, however, drew a sharp reaction from Western nations and the Syrian opposition.
The Syrian National Council, the biggest opposition group, said Russia and China were "responsible for the escalating acts of killing", calling the veto "an irresponsible step that is tantamount to a license to kill with impunity".
UK Foreign Secretary William Hague said the two countries were making a "great mistake", accusing them of "turning their backs on the Arab world".
The BBC's Jim Muir in Beirut says the Russians do seem to be feeling the pressure. They are sending their foreign minister, Sergei Lavrov, to talk to Syrian President Bashar al-Assad in Damascus on Tuesday.
Russia would like to help mediate a political solution, but the opposition does not see the Russians as honest brokers, our correspondent says.
The veto coincided with one of the bloodiest days since protests began last March.
Activists circulated a list of 55 names of people killed after Syrian government forces began to bombard Homs on Friday night. Other activists said the toll was more than 200, though the figure cannot be confirmed.
Homs was one of the first cities to join the uprising and has been repeatedly targeted by government forces.
In continuing violence on Sunday, at least two civilians and nine soldiers were killed in clashes in the north-western province of Idlib, opposition groups said.
Casualty figures are hard to verify as most foreign media are barred from Syria.
Violence was also continuing, particularly in the north, where nine security personnel were reported killed in clashes in Idlib province.