The clashes, which a war monitoring group said had already killed 1000 people, mostly civilians, continued for a fourth day in Assad's coastal heartland.
In a televised speech Ahmed Sharaa, whose Islamist rebel movement toppled Assad in December, accused Assad loyalists and foreign powers that he did not name of trying to foment unrest.
"Today, as we stand at this critical moment, we find ourselves facing a new danger - attempts by remnants of the former regime and their foreign backers to incite new strife and drag our country into a civil war, aiming to divide it and destroy its unity and stability," he said.
The top commander of a Syrian Kurdish armed group, whose forces are in a separate battle with Turkey, had earlier blamed Turkish-backed Islamist factions for some of the most disturbing violence: the reported executions of civilians belonging to Assad's Alawite sect.
Turkey did not immediately respond to the allegation.
Interim president Sharaa's office said it was forming an independent committee to investigate the clashes and killings by both sides.
Syrians have circulated graphic videos of executions by fighters.
Reuters could not immediately verify the videos.
"We will hold accountable, with full decisiveness, anyone who is involved in the bloodshed of civilians, mistreats civilians, exceeds the state's authority or exploits power for personal gain. No one will be above the law," Sharaa added in the video speech after earlier calling for national unity.
A Syrian security source earlier said the pace of fighting had slowed around the cities of Latakia, Jabla and Baniyas while forces searched surrounding mountainous areas where an estimated 5000 pro-Assad insurgents were hiding.
Militants led by Sharaa's Sunni Islamist Hayat Tahrir al-Sham group ousted Assad's government.
Assad fled to Russia, leaving behind some of his closest advisers and supporters, while Sharaa's group led the appointment of an interim government and took over Syria's armed forces.
Assad's overthrow ended decades of dynastic rule by his family marked by severe repression and a devastating civil war that began as a peaceful uprising in 2011.
Since Assad's overthrow, Turkish-backed groups have clashed with Kurdish forces that control much of northeastern Syria.
Israel has separately struck military sites in Syria and is lobbying the US to keep Syrian forces weak, sources have told Reuters.
After the relative calm following the ousting of Assad, violence has spiralled as forces linked to the new Islamist rulers began a crackdown on a growing insurgency from the Alawite sect in the provinces of Latakia and Tartous.
The Syrian Observatory for Human Rights, a UK-based war monitor, said on Saturday more than 1000 people had been killed in the two days of fighting.
It said 745 were civilians, 125 members of the Syrian security forces and 148 fighters loyal to Assad.
Rami Abdulrahman, the head of the observatory, said the civilians included Alawite women and children.
Syrian security sources said more than 300 of their members had been killed in clashes with former army personnel owing allegiance to Assad in co-ordinated attacks and ambushes on their forces that began on Thursday.
Syria's state news agency SANA reported on Sunday that a mass grave had been discovered near Qardaha, Assad's home town, containing the bodies of recently killed security forces.
The attacks spiralled into revenge killings against Alawites when thousands of armed supporters of Syria's new leaders from across the country descended to the coastal areas to support beleaguered forces of the new administration.
Alawites, who belong to an offshoot of Shi'ite Islam, became associated with Assad's wartime atrocities against Syria's mostly Sunni Muslim population because of their religious group, which counted among his most ardent supporters.
The United States and United Nations human rights chief Volker Turk called on Syria's interim leadership to bring the perpetrators to justice.
Damascus authorities have blamed summary executions of dozens of youths and deadly raids on homes in villages and towns inhabited by Syria's once-ruling minority on unruly armed militias who came to help the security forces and have long blamed Assad's supporters for past crimes.
A resident of the town of Qadmous told Reuters people in the town and surrounding villages had fled to nearby fields to protect themselves.
He said a convoy of fighters with tanks, heavy weapons and small drones had burned homes and cars along the main road near his town.
"We don't know how many people are killed yet because they haven't gone home and don't plan to for the next few days," said the resident, who spoke on condition of anonymity for fear of reprisals.
