It comes after loyalist bands defied a ruling by the Parades Commission that only a single drumbeat should be played when passing the church on Donegall Street during a parade on Saturday 25 August.
The ban was imposed after a loyalist band played a song alleged to be sectarian outside the same church on the Twelfth of July.
At the time, parish priest Father Michael Sheehan told UTV the actions of the bandsmen "came across as a sense of saying, we don't care about your church, your building, your people, we'll do what we want".
In an open letter, Millar Farr, the Sovereign Grand Master of the institution, said: "We apologise for any offence to the clergy and parishioners of St Patrick's Church.
He said "the sense of injustice and hurt felt by the members of the Royal Black Institution" was focused on the Parades Commission - not St Patrick's Church.
"Parading is embedded in the DNA of the Protestant community but the Parades Commission has shown an appalling lack of understanding about what that means," he said.
The letter came after three nights of trouble in the Carlisle area of north Belfast, which started after a republican parade on Sunday.
Martin McGuinness said: "The Black Institution showed leadership overnight and I very much welcome that."
Even though it was three terrible nights of trouble a golden opportunity may now present itself which can be developed in the interests of everybody and lead to a final resolution of the whole issue of parading.
The institution's letter called for the abolition of the parades watchdog, whose determinations were described as "irrational and often irresponsible".
"The Institution calls on our politicians to remove the Parades Commission, an unelected quango, and to have it replaced within a democratic framework where decisions will be held accountable to the people."
The order said it hoped to meet the new NI Secretary of State "to repeat our message that the current Parades Commission is a major hindrance to the development of democracy within this country".
The institution has also requested meetings with the leaders of three unionist parties "to enlist their support in bringing about the demise of the Parades Commission", as well as talks with Protestant church leaders.
"I think we are all on the same page," said Peter Robinson after talks at Stormont with PSNI Chief Constable Matt Baggott.
The discussions come ahead of the Ulster Covenant Centenary parade which is expected through north Belfast on 29 September.
I don't think there's any question that action is required to ensure that there's no return of the violence, to ensure that we relieve the parading issue for the end of the month and to ensure that we tackle the wider issue of parading in Northern Ireland.
"I am glad to say we do find common ground in much of what we see the problems being, and the best way in terms of how we process to resolve those issues," he said.
Ulster Unionist leader Mike Nesbitt said the order demonstrated "a much needed spirit of generosity", but added that timing was tight before the Ulster Covenant centenary parade.
He added: "If the will is there, much can be achieved".
Alban Maginness, the SDLP MLA for North Belfast, has urged those involved in the parades dispute to use the space created by the apology to work positively for a respectful solution.
"While the Royal Black Institution's statement does not answer all the outstanding questions or solve all our problems in north Belfast, their apology to the parish is a positive step which helps to create space for reflection," he said.
Meanwhile the order said it has always had "good lines of communication with the Roman Catholic Church" and intend to maintain this "away from the public gaze."
The President of the Methodist Church, Rev. Ken Lindsay, commended the leaders of the order for the move.
Fr Sheehan, who said he appreciated the "sincere Christian spirit" behind the apology" is expected to respond fully to the letter early next week.