Syria has condemned Turkish military action against Kurdish fighters in northern Syria described it as a violation of its sovereignty.
It called on the UN Security Council to take action.
Turkey carried out a second day of shelling on Sunday of Kurdish forces advancing in northern Aleppo province.
Ankara views the Kurdish militia in Syria as allied to the outlawed PKK, which has carried out a decades-long campaign for autonomy in Turkey.
But the United States and others back the Kurdish militia in Syria, the YPG, in its fight against the so-called Islamic State (IS) group.
Syria has accused Turkey of violating its sovereignty by backing “al-Qaeda-linked terrorists” in the north and has warned it has a right to respond.
“Turkish artillery shelling of Syrian territory constitutes direct support to the armed terrorist organisations,” the Syrian government said in letters to the UN Secretary General and the Security Council’s Chairman.
Image copyright AP Image caption The Syrian army aided by Russian airpower is moving ever closer to Aleppo, where conditions are harsh for civilians
The letters accused Turkey of allowing about 100 gunmen – believed to be either “Turkish soldiers or Turkish mercenaries” – to cross into Syria.
“[Syria] will maintain its legitimate right to respond to the Turkish crimes and attacks and to claim compensation for the damage caused.”
France’s foreign ministry has also urged Turkey to end its assault on Kurdish fighters in northern Syria.
Analysis: Mark Lowen, BBC News, Istanbul
Syria’s war hardly needs a new front, but another has now opened up.
Ankara sees the Kurdish PYD party and its YPG militia as terrorist organisations. It’s worried that weapons are seeping over the border from Syria to the PKK, with which Turkey has renewed its armed conflict since a ceasefire broke down last year.
Washington rejects Ankara’s view that the PYD is a terrorist organisation, seeing it instead as an effective ally against so-called Islamic State (IS). That has caused rifts between the two Nato allies and is hampering international peace talks.
Syria is the stage for several proxy wars, and Turkey has just started another, going after its Kurdish nemesis there and vowing to stop the PYD from cementing control of northern Syria.
Russia meanwhile has announced that the US and Russian presidents have agreed in a “frank and business-like” phone call to work more closely to implement the agreement reached in Munich on Thursday, under which world leaders pledged to work towards a cessation of hostilities in Syria within a week.
Russia is an ally of Syrian President Bashar al-Assad and has been carrying out air strikes since September against what it terms “terrorists”.
Russia argues that the “cessation” does not apply to its air strikes, which have tilted the balance of the war in favour of the Syrian government.
A Kremlin statement insisted that Russia remained committed to its battle against IS and “other terrorist organisations” fighting the Syrian government. Some of these groups are deemed moderate by the West.
Turkey’s shelling of the Kurdish militia in Syrian began on Saturday.
The YPG rejects Turkey’s demand to leave areas it has seized, saying Islamists would return if it left.
Almost five years of civil war in Syria have led to the deaths of more than 250,000 people. More than 11 million people have been displaced.