CAIRO—Egypt’s military leaders said they would push ahead with parliamentary elections beginning Monday, while refusing to give in to swelling protests calling for a quicker transition to civilian rule.

The elections—the country’s first parliamentary vote since former President Hosni Mubarak’s ouster in February—could stand as a turning point in Egypt’s transition from six decades of dictatorship to civilian democratic rule.

Associated PressA woman reached into a barricade set up on Thursday by Egyptian troops at Cairo’s Tahrir Square.

But some activists and political party officials have called for a delay in the vote, saying that if the unusually complicated election process is marred by continued violence, low voter turnout or suspect results, it could prolong military rule and further destabilize the country.

Abdel Moez Ibrahim, head of the Supreme Electoral Committee, said the country’s independent judges would monitor the vote. “We are ready to hold elections under any circumstances,” he said Thursday.

With Egypt’s economy in peril—Standard & Poor’s cut the country’s credit rating to single-B-plus from double-B-minus on Thursday—popular frustration looks primed to spread, without a government that has the popular mandate to make tough decisions.

Another major demonstration was planned for Friday, and protesters said they would continue their fight until the military hands power to a civilian government.

After five straight days of escalating bloodshed between protesters and security forces that left 38 dead, thousands injured and hundreds arrested, a fragile calm held in Cairo on Thursday. A dawn truce negotiated by Muslim clerics took effect and the military began erecting concrete barriers around portions of Tahrir Square, the center of demonstrations, to prevent protesters from marching on the Ministry of Interior.

The military on Thursday apologized for the deaths of protesters and said it would investigate those responsible, in a bid to appease activists. But military leaders continued to insist security forces were acting in self-defense and called on citizens to step in as vigilantes to arrest troublemakers. A similar call for citizen action during a protest in October led to violent clashes that left 27 people dead.

Clashes between Egyptian security forces and protesters stop for the first time in Cairo since Saturday, though violence continues elsewhere in the country. (Video: Reuters / Photo: Getty Images)

Among those arrested in the past week were prominent Egyptian-American columnist Mona Eltahawy, who said via Twitter that she was sexually assaulted, beaten, and blindfolded during her 12-hour detention by police.

A spokesman for the Ministry of Interior said he had no information regarding Ms. Eltahawy’s arrest or detention. As for the accusations of sexual assault, he said the ministry denied such accusations “overall and specifically.”

Egyptian-American filmmaker Jehane Noujaim was arrested and remained in custody Thursday evening, according to a statement by Ms. Noujaim’s supporters.

Even before the latest bout of protests and clashes erupted on Nov. 18, many Egyptians feared that the elections could bring violence. Many activists and individual party officials are calling for the elections to be postponed for at least a few weeks until calm returns.

Polling will be divided by geographical area into three rounds, concluding in early January. Cairo and Alexandria, two cities that have seen the most violence in recent days, are among those voting on Monday.

The ballot was only recently finalized and is extremely complex, with scores of new parties and candidates, adding to the prospect that Egyptians—long accustomed to an election process that kept Mr. Mubarak and his party in power—will fight the results.

Many Egyptians doubt the military and police will be able to muster sufficient force to secure polling stations on voting day. The recent violence between protesters and police has further eroded public trust in security services, raising concerns that voters will stay away from the polls—despite a 500 Egyptian pound ($85) fine on anyone who registers but fails to vote.

Gen. Sami Enan, a member of the ruling military council, told political party leaders on Tuesday that the Interior Minister had said he lacked sufficient police to provide security on Election Day, according to a party leader. Army commanders say they can backstop any police shortfalls in poll security.

Egypt has a long history of election-related bloodshed, with candidates often relying on hired thugs to intimidate rival voters, and frequent clashes between police and voters.Many Egyptians, accustomed to the practice, expect violence to continue in this election.

Most political parties have suspended or drastically curtailed campaigning to show solidarity with protesters, though parties have been caught between youth cadres demanding an immediate power transfer and an older generation who appear uncomfortable openly challenging the powerful military.

The country’s largest opposition group, the Muslim Brotherhood, one of the most vocal supporters of pushing ahead with elections as scheduled, has seen divisions within its ranks over whether to support protesters or continue dealing with the military.

At least 150 members of its youth ranks have left the organization in recent days, in protest against the group’s negotiations with military leaders and over its refusal to back protesters in Cairo’s Tahrir Square.

A senior Muslim Brotherhood leader, Mohammed al-Baltagi, has publicly criticized the group’s position, exposing an exceedingly rare public rift within the disciplined organization.

Influential Muslim cleric Yussuf al-Qaradawi, who is revered by Muslim Brotherhood rank and file and was once offered but declined the group’s top leadership position, arrived in Cairo on Wednesday night, likely to help heal the rift, a mediating role he has played for the group in the past, according to those familiar with the organization.

—Matt Bradley contributed to this article. Write to Tamer El-Ghobashy at tamer.el-ghobashy@wsj.com and Charles Levinson at charles.levinson@wsj.com