“I believe the right thing is to stand on this road,” she said. “What we can’t see on this road is the return of a triumphalist march that railroads over the rights of this community.”
Earlier, Northern Ireland’s top police officer Jon Boutcher said the force’s “principal objective” is to police the decision to allow a parade down the Garvaghy Road in Portadown, but added this would not be done “at any cost”.
He said the overriding responsibility of police was to keep people safe.
There has been more than three days of stalemate over the Orange Order parade.
O’Neill said on Wednesday that political unionism supporting the Drumcree parade “goes right to the heart of power-sharing”.
“I think it does put into question all of the apparatus of the Good Friday Agreement, including the institutions,” she added.
“All I can say is that I am committed to trying to make them work, even though they are a very challenging arrangement.
“I’ve turned up this week to try to make it work, I will continue to do so, as for other’s actions, I can’t speak for them.”
She said Sinn Féin would continue to play its role across the assembly and the executive.
Unionist politicians have been highly critical of nationalist politicians who attended Sunday’s protest.
The leader of the Democratic Unionist Party (DUP) Gavin Robinson said those in attendance had broken the law.
He met with Boutcher at PSNI headquarters on Wednesday afternoon and speaking afterwards he said the “law applies equally and to everyone”.
“We have given him our expectation that within days, if not a week, we need to see the positive out workings of that,” he added.
“The rule of law pertains and sustains in our society, that it is not going to be set aside for anyone, no matter who they are.”
