South Sudan's President Salva Kiir says the actions of neighboring Sudan amount to a declaration of war.
His remarks Tuesday came during talks with Chinese President Hu Jintao in Beijing. Kiir is in China on a five-day trip that will include opening a new embassy for South Sudan.
The U.S., China and other world powers have called on Sudan and South Sudan to stop fighting and resume peace talks.
Sudan's president, Omar al-Bashir, said Monday there will be no further talks and that the south only understands the language of "guns and ammunition."
South Sudanese officials said Tuesday that Sudanese warplanes dropped bombs overnight in Panakwach.
On Monday, officials and witnesses in South Sudan said Sudan bombed inside the South's Unity state, killing at least three people, including one boy. The bombs struck in and around the border town of Bentiu.
U.N. Secretary-General Ban Ki-moon condemned the Sudanese bombings and called on the government in Khartoum to cease all hostilities immediately.
The United States also condemned what it termed Sudan's "military incursion." State Department spokeswoman Victoria Nuland said the U.S. recognizes the South's right to self-defense, but urged Juba to "exercise restraint" in its response to the bombardments.
The deputy head of South Sudan's military intelligence, Mac Paul, accused Sudan of "continuous provocations" since the south withdrew its forces from the Heglig region in recent days.
South Sudan says it made an orderly and voluntary pullout from Heglig, following international pressure to withdraw. Sudan says it retook the area by force.
The two countries have been unable to resolve disputes over borders, oil and citizenship stemming from the south's independence last July.
The Sudan and South Sudan previously fought a 21-year civil war that killed more than two million people. The war ended with a 2005 peace agreement that included an independence referendum for the South.
Some information for this report was provided by AP and AFP.