The federal government's decision in April to dismantle regional forces established by some states prompted violent protests in Amhara, with Amhara nationalists claiming the move would weaken Ethiopia's second most populous region.
Late Wednesday, Demeke said that "the security problems that are seen in different areas of Amhara region are becoming worrisome."
"We are at a historical time where we should be mindful of the fact 'If you don't have peace you will lose everything'," he posted on Facebook, without identifying the parties involved in the clashes.
Ethiopian army spokesman Getnet Adane told a press conference this week that fighters claiming to be from the local militia Fano were responsible for the violence.
"We will take measures against those who in any way attacked our army or facilitated the attack on our army," he said, denouncing fighters "who trade on the name of any force," including Fano.
Britain's Foreign Office warned its citizens against traveling to certain parts of Amhara, citing "increased violence in these areas characterized by Fano taking control of these areas."
"Most recently Lalibela Airport has been taken over by Fano militias," it said, referring to the tourist town which is home to 12th and 13th century rock-cut churches listed as UNESCO World Heritage Sites.
No Ethiopian media has reported on any incidents in Lalibela, but the Spanish Embassy in Addis Ababa on Tuesday also urged its nationals not to travel to Amhara, citing "instability" in the region.
"Spaniards who are in Lalibela are advised not to leave their hotels or homes and contact the Embassy in Addis Ababa," it said on X, the platform formerly known as Twitter.
Amhara's regional forces and local militias backed federal troops in their two-year war against Tigrayan rebels, until a peace deal was signed in November 2022, angering Amhara nationalists.
Despite the peace agreement, Amhara special forces and Fano fighters continue to control Western Tigray, an area claimed by both the Amhara and Tigrayans.
Prime Minister Abiy Ahmed said in April that the decision to disband regional forces and integrate them into the national army or regional police would bolster multi-ethnic Ethiopia's "unity."