Loyalist protesters who gathered outside Carnmoney Cemetery in County Antrim threatened to desecrate Catholic graves, a local priest has said.
Police were on duty in the cemetery
It happened during a blessing ceremony on Sunday, which has been the focus of previous loyalist paramilitary threats.
Parish priest Father Dan Whyte said people were "very upset".
"I was disappointed when I learned that later there was a protest that degenerated into a noisy chorus of sectarian verbal assault," he said.
He also said that "verbal threats of grave desecration" had been made.
Fr Whyte said that he had hoped the situation would have returned to normal, particularly since the ceremony had been delayed from September to allow "temperatures to cool off".
He said that Catholics and Protestants should unite to oppose such attitudes.
The DUP mayor of Newtownabbey, William de Courcy, said he was disappointed the protest took place, but that he could not say people did not have a right to protest.
"I think it's sad that they have to do it at a time of such sadness and such sacredness of the Catholic community," he said.
Sinn Fein councillor Briege Meehan condemned the protests.
"Once again the Catholic community in Newtownabbey have been subjected to naked sectarian hatred and bigotry in its most vile form as they paid devotion to their deceased loved ones in a dignified and non-triumphalist manner," she said.
She called on unionist politicians to ensure the protests do not happen again.
Fr Whyte said threats to desecrate graves were made
Two protests were held at the cemetery in Newtownabbey where Catholics had gathered for an annual blessing of their graves ceremony.
Police said an initial demonstration was peaceful. A second protest was held later in the day and the O'Neill Road was closed.
The ceremony was delayed, but was able to proceed after a short time.
Speaking after Sunday's ceremony, before learning that threats had been made, Fr Whyte, of St Bernard's Church, Glengormley, said: "Our objection was that in previous years, the protest was a riotous protest, the purpose of which was to prevent us from saying our prayers.
"That wasn't the case today.
"I'm thinking perhaps that the bad old days of Carnmoney Cemetery Sunday are behind us now."
Traffic diversions were put in place outside the cemetery for the initial demonstration.
It is understood that community representatives were present and helped police ensure that the protest passed without incident.
In September, the graves ceremony was postponed by Father Whyte following widespread violence across Northern Ireland.
Carnmoney graveyard has been attacked on several occasions and headstones have been smashed.