Senegal's players walked off the pitch in protest at a VAR-awarded penalty before returning to beat hosts Morocco 1-0 after extra time on Sunday.
Groups of Senegalese fans clashed with Moroccan security as they tried to enter the pitch following the penalty decision.
Senegal has long been one of Morocco's closest allies in Africa, where Moroccan firms and banks have expanded investments in recent years, strengthening the kingdom's diplomatic influence.
Hours after the final, social media in both countries was under the spell of tension.
The final had been marred by "regrettable incidents" and "deplorable behaviour," the king said in a palace statement.
"Once the passion and emotions have subsided, inter‑African fraternity will naturally prevail," he said.
"Nothing can undermine the close ties forged over the centuries between our African peoples, nor the fruitful cooperation built with countries across the continent and strengthened by increasingly ambitious partnerships," he added.
The tournament had also showcased Morocco's development and represented "a success for all of Africa," he said.
The Confederation of African Football (CAF) said it would take "appropriate action" after reviewing the match incidents, while FIFA President Gianni Infantino condemned the behaviour of Senegal's players and members of the coaching staff.