President Salva Kiir of South Sudan says he won't withdraw troops from the Heglig oilfield on its disputed border with Sudan.
In parliament on Thursday, Mr Kiir said he did not want war with Khartoum, but said the country had to act in self-defence.
''I always say we will not take the people of South Sudan back to war, but if we are being aggressed (attacked) like this we will have to defend ourselves,'' he said.
''I am appealing to the citizens of the Republic of Sudan, especially the mothers, not to allow their children to be dragged into a meaningless war,'' he added.
Meanwhile, Sudanese aircraft are reported to have attacked Unity state in the South.
The fighting is the worst since South Sudan gained independence last July.
On Wednesday, Sudan admitted that South Sudan had seized the town of Heglig, which is home to facilities that account for around half of Sudan's oil production.
The area is internationally recognised as Sudanese territory, but South Sudan disputes this.
South Sudan says it took Heglig in order to prevent the Sudanese army organising attacks from there.
On Wednesday, the African Union said it was deeply alarmed by the clashes, and called on both sides to exercise the utmost restraint.
In Geneva, UN Secretary General Ban Ki-moon said he was gravely concerned about the situation and urged both governments to "cease hostilities immediately"