The proposals, which include the removal of the crown from the prison symbol and prison uniforms, form part of fundamental reform outlined by the former Chief Inspector of Prisons, Dame Anne Owers, and her review team.
They were detailed by Mr Ford after the issue was raised by TUV leader Jim Allister in the Assembly on Monday.
Peter Robinson later made it clear that he would take the ultimate step in quitting his job and triggering an election, if the symbolic changes - which have been suggested by the Prison Service - were enforced.
"Not on my watch," the DUP leader told party colleagues.
If we are unable to block it in the Assembly, I will resign my position in order to force an election and allow the electorate to decide the issue.
First Minister Peter Robinson
Speaking to UTV, Mr Robinson added: "This came completely out of the blue - David Ford has never mentioned it to me on any occasion.
"So let's be very clear, he is badly advised if he thinks that this is something that can be done in an operational manner."
Mr Ford, who met with Mr Robinson on Tuesday afternoon, insists that no decisions have been taken.
"As everyone will be aware, we are currently engaged in an extensive programme of reform within the prison system," the justice minister said.
"The issue of symbol is clearly part of the issue of what is under consideration but there are no specific plans or proposals on my desk at present," he told UTV Live.
"Clearly issues of controversy like the symbols would be issues which would have to be discussed by other executive ministers."
Mr Ford described his meeting with Mr Robinson as "sensible and reasoned".
"It was looking at the way forward and I had a similar constructive discussion with Martin McGuinness as well this afternoon," he said.
I cannot give any assurance that the symbols will change or that they will not change.
Justice Minister David Ford
"The challenge for all of us collectively is to work together to deliver the level of prison reform that our community needs," Mr Ford said.
DUP Justice Committee Chairman Paul Givan said he welcomed the clarification "that any change to the Northern Ireland Prison Service symbols or the title used by our prisons would be a matter for the Executive".
"As the largest Party in the Executive, the DUP would veto any attempt to remove Her Majesty from our Prisons' titles," he said.
Earlier Sinn Féin Deputy First Minister Martin McGuinness urged Mr Robinson to "calm down".
He said: "Obviously the whole issue of prison reform is of vital importance.
"That's one of the reasons why Anne Owers and her team were involved in looking at the prison system and are coming forward with very important proposals which will fundamentally change the whole system.
"So let's be sensible about this. Let's be very calm about how we deal with this and I think there should be no knee-jerk reactions."
SDLP member of the Justice committee Colum Eastwood said the First Minister "should focus on the substantive issues".
"Political leaders should not be afforded the luxury at this time of financial turbulence to indulge in controversies surrounding symbolism," he said.
There are big challenges out there and threatening resignations and elections is not what people voted for last May and it is not helpful.
Raymond McCartney, Sinn Féin Assembly Group leader
The chairman of the Prison Officers' Association, Finlay Spratt, has hit out over the proposals and claimed that they would be "a sop to terrorism".
Meanwhile Mr Allister, who tabled a no-confidence motion in the justice minister, described the planned changes as "a calculated insult" to prison officers.
"David Ford's attempt to bring such radical change to the Prison Service without so much as consulting the Assembly shows that he is unfit for office," the North Antrim MLA said.
"If Peter Robinson really is so exercised about Mr Ford's plans, doubtless the DUP will back this (no confidence) proposal," he said.
UUP leader Tom Elliott said he was "outraged" at the proposals and described them as "an insult to the bravery and suffering of the men and women of the Northern Ireland Prison Service".
Responded to Mr Robinson's threat to resign, he added: "The Ulster Unionist Party has long believed that David Ford was little more than a DUP puppet, and the reaction of Mr Robinson to the first decision by Mr Ford that the DUP disagrees with, would tend to confirm those suspicions."