Much of the heaviest fighting has raged in the capital Khartoum and in Darfur, near the border with Chad, since the conflict erupted on April 15.
Minawi, a former rebel leader, has voiced support for the national army in its battle against the paramilitary Rapid Support Forces.
"There are many who do not wish for the safety or rights of citizens and deliberately sabotage national institutions," he wrote on social media. "I call on all our honorable citizens, the people of Darfur, old and young, men and women, to take up arms to protect their property."
Darfur has already suffered decades of turmoil that has left hundreds of thousands dead, more than two million displaced and the region flooded with weapons.
The war there, which began in 2003, saw Sudan's then president Omar al-Bashir unleash the feared Janjaweed militia to crush a rebellion among ethnic minority groups.
The Rapid Support Forces, commanded by Mohamed Hamdan Daglo and now at war with the Sudanese army, traces its origins to the Janjaweed.
Darfur has seen some of the worst of current fighting, with hundreds of civilians killed, markets burned and rampant looting of health and aid facilities.
Tens of thousands of Sudanese have fled across the border into Chad as concerns rise about the militarization of those who remain.
Sudanese democracy activist and author Raga Makawi said there is "a real risk of people who were in the past part of nonviolent movements now considering the right to bear arms in order to protect themselves".
The Small Arms Survey, a Geneva-based research project, calculated there were 6.6 guns for every 100 people in Sudan in 2017.
The United Nations had already warned that civilians were being armed in the fighting before Minawi issued his call to arms.
The army, under chief Abdel Fattah al-Burhan, has also sought to reinforce its ranks. On Friday the defense ministry called on "army pensioners" and reservists to head to command units.
The fighting across Sudan has killed more than 1,800 people, according to the Armed Conflict Location and Event Data Project.
The United Nations says more than a million people have been displaced within Sudan, in addition to 319,000 who have fled to neighboring countries.
A one-week ceasefire brokered by the United States and Saudi Arabia and marred by breaches is due to expire Monday night.