Clinton also urged regime troops involved in an 11-month brutal crackdown on dissent to renounce violence against civilians.
"We are appealing to the members of the Syrian army to put their country first," Clinton said in Morocco.
"The longer you support the regime campaign of violence against your brothers and sisters, the more it will stain your honour," she said addressing regime supporters including the military.
The text of the new constitutions ends the legal basis for the five-decade stranglehold on power of the ruling Baath party but leaves huge powers in the hands of President Bashar al-Assad.
The opposition says the changes are cosmetic after nearly a year of repression by Assad's security forces that human rights groups say have left more than 7600 people dead.
Drawn up by a committee of 29 people appointed by Assad, the new charter would drop the highly controversial Article 8 in the existing charter, which states that the Baath party is "the head of state and society".
The new political system would be based on "pluralism" and ban the formation of parties on religious lines.
But under the new charter, the president would retain broad powers, such as naming the prime minister and government and, in some cases, could veto legislation.
Article 88 states that the president can be elected for two seven-year terms, but Article 155 says these conditions only take effect after the next election for a head of state, set for 2014.
Assad could therefore theoretically stay at the helm for another 16 years.
The opposition Syrian National Council piled the pressure on Assad by inviting the Alawite community from which he hails to join ranks to build a new Syria.
"We are determined to close national ranks and the first sign of this unity is to extend our hands to our Alawite brothers in order to build a country of nation governed by citizenship and the rule of law," it said.
The SNC accused the regime of trying to set religious communities against each other and stressed that Alawites - who represent 12 per cent of the mostly Sunni population of 22 million - will always be an "important component of Syria".
In Homs - under assault by regime forces for more than three weeks - shelling resumed of the rebel district of Baba Amr, dashing Red Cross hopes of a lull to allow the evacuation of two wounded Western journalists.
"The evacuation will not happen Sunday because it is dangerous to send ambulances at night," said ICRC spokesman Saleh Dabbakeh. "It will take place most likely on Monday," he added in Damascus.
The wife of a British photographer injured in an army bombardment of Homs, meanwhile, urged London to rescue her husband despite the government deeming the mission too dangerous.
British photographer Paul Conroy and French reporter Edith Bouvier were wounded in the attack on Wednesday which claimed the life of American war correspondent Marie Colvin and French photographer Remi Ochlik.
France's Interior Minister Claude Gueant warned it was "medically urgent" to get Bouvier out of the besieged district.
Syrian state television aired live footage from a number of polling stations around the country and reported that "large number of voters" had turned out.
In the southwestern city of Sweida, heartland of the Druze minority, a woman said she voted without hesitation in favour of the new constitution, although she said it was not enough.
"We're not stupid. We know that the new text does not meet the aspirations of the people, but voting is a message to support stability and reject the civil war that threatens our country," she said.
In Homs, no voting appeared to have taken place, activist Hadi Abdullah told AFP after touring parts of the city where rebels are active.
Foreign journalists have very limited freedom of movement in Syria because of stringent restrictions imposed by the authorities.
Assad unveiled the proposed new national charter earlier this month, in the latest step in what he says is a cautious process of reform.
- AAP
