But Ms O’Neill accused the unionist party of “punishing the electorate” while Alliance Party leader Naomi Long said it had been a “shameful day” for the DUP.
The failure to elect a Speaker leaves the Stormont Assembly unable to function.
The DUP is also refusing to nominate for the position of deputy first minister, which prevents the forming of a new executive, as part of its protest against the protocol.
The 90 MLAs met for the first time in the Stormont chamber on Friday after last week’s Northern Ireland Assembly election saw Sinn Fein emerge as the largest party for the first time.
The first order of business was for MLAs to sign the roll of membership before an attempt was made to elect a Speaker.
Unionists oppose the post-Brexit treaty because of the economic barriers it creates between Northern Ireland and the rest of the UK.
Outlining his reasons for blocking the election, Sir Jeffrey said: “As I have made clear this morning we have taken the decision not at this stage to support the election of a speaker.
“I believe that we need to send a very clear message to the European Union and to our Government that we are serious about getting this protocol sorted out.
“Because of the harm it is doing, undermining political stability, damaging the agreements that have formed the basis of political progress made in Northern Ireland, to our economy, contributing to the cost-of-living crisis, this matter needs to be dealt with.
“While others sit on their hands we are not prepared to do that. We need decisive action taken by the Government.”
He added: “The choice is clear: if the European Union is serious about protecting the political institutions and the Belfast agreement, and its successor agreements, then they know what to do.
“Equally the same message is there for our own Government as well. The ball is firmly at the foot of the Government.”
Sinn Fein Stormont leader Ms O’Neill responded by saying the DUP has “punished the electorate” by boycotting the election of a speaker of the Northern Ireland Assembly, and that “the public deserve better”.
She added that the DUP’s action to boycott the election of an Assembly speaker “isn’t tolerable, it isn’t acceptable, it isn’t good enough”.
Ms O’Neill confirmed that the Prime Minister would be visit Northern Ireland on Monday.
Alliance Party leader Mrs Long said: “While this is a sad day for the people of Northern Ireland, it is a shameful day for the DUP.”
As the largest party, the 27 Sinn Fein MLAs took their position on the benches on the right-hand side of the Speaker’s chair for the first time.
It is entitled to nominate its Stormont leader Michelle O’Neill as first minister, but she will not be able to take up the role without the DUP nominating a deputy first minister.
Under the rules of the devolved power-sharing administration, both roles are equal and one cannot be in office without the other.
Since last week’s election, Ms O’Neill has repeatedly called for the DUP to re-enter the executive so it can begin to tackle challenges such as the cost-of-living crisis and spiralling hospital waiting lists.
The Stormont sitting comes amid increased tensions between the government in Westminster and the EU over the working of the protocol, which forms part of the UK’s Brexit agreement.
It comes after Foreign Secretary Liz Truss said the UK will have “no choice but to act” if the EU does not show enough “flexibility” over post-Brexit checks on goods going from Britain to Northern Ireland.