File photo: The opening session of the 33rd African Union (AU) Summit held in Addis Ababa in February. AP
The comments by Moussa Faki Mahamat follow claims by Khartoum that the Ethiopian army had executed seven Sudanese soldiers and a civilian in an incident in a volatile border region last week -- allegations denied by Addis Ababa.
Faki "is following with deep concern the escalating military tension between the Federal Democratic Republic of Ethiopia and the Republic of Sudan and deeply regrets the loss of life at their common border," the AU said in a statement.
"The chairperson appeals for complete refrain from any military action whatever its origin and calls for dialogue between the two brotherly countries to solve any dispute."
Sudan accused Ethiopia of capturing the soldiers on June 22 in Al-Fashaqa, a fertile strip at the centre of a bitter border dispute between the two neighbours.
Khartoum recalled its ambassador to Addis Ababa on Monday and vowed to lodge a complaint with the UN Security Council and regional organisations.
Addis Ababa said Tuesday that Sudanese forces had crossed into Ethiopian territory and the casualties resulted from a skirmish with a local militia, denying its soldiers were in the area at the time.
The Sudan Tribune newspaper reported that the Sudanese army had launched an attack Tuesday on Ethiopian troops in the Al-Fashaqa area but this was denied by army spokesman Nabil Abdalla.
"We have not attacked anyone and we will not and we are not planning that. But we will not allow any armed force from another country that wants to cross our international border. It's our right legally to deal with it," he told AFP in Khartoum.
Relations between Khartoum and Addis Ababa have soured over Al-Fashaqa, which is close to Ethiopia's restive Tigray region.
Al-Fashaqa has long been cultivated by Ethiopian farmers but claimed by Sudan, and the dispute has sparked sporadic clashes between the Sudanese and Ethiopian sides, some fatal.
The rift over Al-Fashaqa feeds into wider tensions over land and water between the neighbours, particularly stoked by Ethiopia's mega dam on the Blue Nile.
Sudan and Egypt, both downstream countries, have been opposed to the Great Ethiopian Renaissance Dam and pushed for an agreement on the filling of its reservoir and the dam's operations.