The church had called for "peaceful" memorial rallies on February 16 to mark a 1992 "March of Christians" that was violently put down by the regime of the erstwhile dictator Mobutu Sese Seko.
"This initiative drew contrasting and disproportionate reactions from our countrymen," said Monsignor Nicolas Djomo, the head of the National Episcopal Conference of the Congo (CENCO), in a letter to priests across the country.
The letter, seen by AFP on Wednesday, said the "Holy See strongly recommended the suspension of initiatives which could be manipulated for political ends," said the letter, written on January 5.
President Joseph Kabila has called for a "national dialogue" to help ensure "peaceful elections" due this year, but the opposition says this is a strategy to get round the constitution and stand for a third elected five-year term.
On Saturday, a new anti-Kabila grouping called the Front Citoyen 2016 grouping political parties and associations had said it would join February's memorial marches
And on Tuesday, the ruling party called for a peace march on the very same day of the memorial rallies.
Djomo said the church's decision also aimed to avoid "possible clashes" during the march as there had been indications that there could be counter-rallies.
"Being conscious that our initiative runs the risk of being hijacked and of possible clashes... it seems wise to defer this march."
Kabila was first asked to take power in 2001 at the age of 29 to replace his murdered father, Laurent Desire-Kabila, during the Second Congo War, also known as the Great Africa War.
The conflict embroiled more than half a dozen countries until 2003 -- partly because others wanted a stake in the DRC's fabulous mineral wealth -- and is reported to have left more than three million people dead.
Kabila took up his first elected term in 2006, under a new UN-supervised constitution which provided for two five-year mandates in the vast nation of some 81 million people.
Eighty percent of the population are estimated to be Christians of different denominations, half of them Roman Catholics.