A spokesman for the U.N. Stabilization Mission said the organization had recorded six deaths, dozens of injuries and more than 240 arrests in Kinshasa when demonstrations were staged after Masses Jan. 21. The church's lay coordinating committee organized the demonstrations.
U.N. radio in Congo reported coalition parties criticized the church for organizing the protests.
Agence France-Presse reported a 24-year-old female religious novice had been killed when police fired on St. Francis de Sales Church in the capital's Kintambo suburb.
It added that similar violence had erupted outside Notre Dame Cathedral and in many of the city's 160 Catholic parishes, as protesters, accompanied by clergy, waved crucifixes and rosaries.
U.N. radio in Congo reported coalition parties criticized the church for organizing the protests and for working "against the interests of the DRC."
The Catholic Church makes up half the 67.5 million inhabitants of Congo and has pressed Kabila to step down since his second and final term expired more than a year ago.
A church-brokered accord in December 2016 allowed the president to stay in office, alongside an opposition head of government, pending elections by the end of 2017.
However, in November, Congo's Electoral Commission said the ballot would be postponed till Dec. 23, 2018.
Earlier in January, the Congolese bishops' conference condemned "violent and bloody repression" of similar protests Dec. 31, in which eight people were killed. The bishops also demanded action against those who "deliberately profaned churches and holy places."
Speaking Jan. 21 during his visit Lima, Peru, Pope Francis urged Congolese leaders to "do everything possible to prevent further violence and seek solutions to the common good." He led a minute's silence for victims.
On Jan. 22, the Association of Bishops' Conferences of Central Africa pledged solidarity with Catholic efforts "to achieve a state of law." Muslim and Protestant leaders had backed the Catholic demonstrations in weekend statements and urged officials not to use force.