Egypt, Eritrea, Somalia and Sudan, which have tense relations with Ethiopia, have called for the settlement of regional disputes through "peaceful means", the Egyptian presidency said in a statement quoted by Agence France-Presse.
The four countries participated in a mini-summit in El Alamein in northern Egypt amid growing concerns that the violence in Ethiopia could spread beyond its borders.
The meeting was attended by Egyptian President Abdel Fattah al-Sisi, Eritrean President Isaias Afwerki, Somali President Hassan Sheikh Mohamud and Sudanese army chief Abdel Fattah al-Burhan.
The four leaders stressed the importance of "respect for the sovereignty, unity and territorial integrity of states, support for their national institutions, rejection of any interference in their internal affairs and settlement of disputes through peaceful means", the statement said. A joint statement after the meeting also stressed the importance of respecting international law governing transboundary water flows, especially the Nile River.
The four countries also expressed "their complete rejection" of any unilateral measures that could harm the water rights of riparian countries.
The declaration also reaffirmed support for freedom of navigation and stressed that the management of the Red Sea is the "exclusive responsibility of the riparian states".
Relations between the four countries and Ethiopia have been strained for years.
The Grand Ethiopian Renaissance Dam, built by Ethiopia on the Nile River, has been a source of ongoing tension with Egypt for more than a decade. Cairo fears a reduction in its water resources and calls the project an "existential threat".
Egypt and Sudan have been pushing for an agreement for years to regulate the operation of the dam.
Somalia and Ethiopia also have traditionally had tense relations, which have worsened since Addis Ababa's rapprochement with the self-proclaimed Republic of Somaliland, which seceded from Somalia in 1991.
Earlier this month, Cairo and Addis Ababa expelled members of the other side's diplomatic missions. This came days after Ethiopia accused Egypt, Sudan and Eritrea of supporting rebels in Ethiopia's northern Tigray region, where renewed fighting has raised fears of a return to civil war.
The three countries have rejected the accusations.
Last week, Ethiopia and Eritrea also severed diplomatic relations, accusing each other of fomenting internal conflicts in the other country.