DUP leader Arlene Foster has ruled out a border poll saying the right circumstances do not exist.
The first minister tweeted: "Irrespective of the view in Dublin or Brussels, a border poll can only be called by SoS if it appears likely to secure a majority in Northern Ireland. No such circumstances exist in Northern Ireland.
"2001 GE 42% for United Ireland.
"2019 GE 38% for United Ireland."
With all 160 seats filled as counting concluded overnight, Sinn Féin now hold 37 seats, one less than Fianna Fáil on 38. Outgoing taoiseach Leo Varadkar's Fine Gael now have 35 seats.
Sinn Féin leader Mary Lou McDonald said she will be asking the European Union to support Irish reunification if she is part of the next government.
Speaking to BBC’s Newsnight in Dublin, Ms McDonald said even before this election the country was heading towards a border poll.
“You have Brexit, you have changing demographics, you have the fact that the unionist majority has been lost in the north over the last number of elections, so that is the direction of travel,” she said.
Ms McDonald said she thinks the European Union needs "to take a stand in respect of Ireland in the say way that it supported the reunification of Germany, in the same way that it has a position on Cyprus, for example, and a positive approach to the reunification of that country".
Ms Foster insisted that Sinn Féin's success in the Republic's general election was a "protest vote" over housing and health issues and not a reflection on support for a united Ireland.
The DUP leader said Irish unity and even Brexit had been dwarfed by other issues.
"What is very clear is that Brexit, which we were told is the issue of a generation, I think was registering at 1 per cent and even on a border poll the height of it is 57 per cent and this is the Republic of Ireland we are talking about," she said.
Read more: Mary Lou McDonald - 'I may well be the next taoiseach'
"I think we should realise that this has been about domestic policy and the fact that it was viewed that the government had failed on those issues and people wanted a change."
Mrs Foster acknowledged that young voters had not been affected by Sinn Féin's historic links to the IRA.
Irrespective of the view in Dublin or Brussels, a border poll can only be called by SoS if it appears likely to secure a majority in Northern Ireland.
— Arlene Foster (@DUPleader) February 11, 2020
No such circumstances exist in Northern Ireland.
?? 2001 GE 42% for United Ireland
?? 2019 GE 38% for United Ireland
"It appears that younger voters are going to Sinn Féin because they are looking at them as the party that gives them something different than Fianna Fáil or Fine Gael and that's the reason why they voted for Sinn Féin. It's a protest vote, I think we should recognise that," she told the BBC.
The first minister said talk of a possible Sinn Féin foreign minister, who could help influence the European Union's post-Brexit trade talks with the UK, was premature.
"We will of course deal with whoever is the minister for foreign affairs, but we should not forget that the constitutional position of Northern Ireland is within the UK, that's why the person who comes from Dublin is the minister for foreign affairs, because we are in a different jurisdiction," she said.
Yesterday, former taoiseach Bertie Ahern said a border poll appeared "inevitable" within the next decade following Sinn Féin's electoral success.
The former Fianna Fáil leader said: "I think a border poll is inevitable. If you ask me when that is, I think it's probably five years off at least.
"I think the Sinn Féin position is it should be held within five years. So let's say we are talking about it being five years off, I do not think you are going to get the circumstances where it would become a condition of government that it has to be held in the short term.
"But it will be inevitable over this decade."
On BBC Radio 4's World at One he warned against rushing towards a poll on reunification before extensive preparatory work had been completed.
"It can only be done when the preparation is done, when the case is made, when it has been well explained, when people know the outcome," he said.
"To try and push it or rush it, which some people will probably try to do, would be a grave mistake."
Deputy First Minister Michelle O'Neill said it was a "good day".
Asked how far up the agenda will preparation for a united Ireland be, Ms O'Neill said: "We said we want to negotiate a programme for government that delivers on what we said to the electorate throughout this election campaign.
"We have provided solutions in terms of the health crisis and the housing crisis so we will want to work with others to establish a programme for government that reflects how we can deliver upon those promises.
"But obviously we are Irish republicans, there is no secret of that, and clearly we will have asks in terms of the republican project, clearly we will have asks in terms of the kind of society we want to build.
"But obviously at the end of the day we have said there should be unity referendum within five years so that's the position we will take into negotiations."
Asked does she believe Ireland has moved closer to unification, she said: "Yes, I absolutely do. We have said things are going in this direction for some time, I think for a number of reasons not least the fact that the unionist majority, the very basis on which this northern state was founded, is gone and has been gone over the last number of elections.
"But I think this is a seismic election shift in the 26 counties, Sinn Féin are the largest party in the 26 counties, I think that is significant in itself."
Speaking at Stormont, the Sinn Féin vice president commented: "Today is a good day on the back of what is a game changer election for Sinn Féin in the 26 counties.
"The people have spoken in large numbers, they have thrown their support behind Sinn Féin. I think that our message of change has very much resonated with the voters. They were tired of the status quo, they were tired of the establishment parties and their arrogance, they were tired of the fact they have been failed on the housing and health front.
"We're very grateful for all those people who came out and voted in such large numbers for the Sinn Féin team.
"We now are involved in the next stage of that which is wait until the votes are all counted towards the end of today but Mary-Lou has already reached out to the other smaller parties who also had a message of change to look about how we can form a government that actually reflects on the needs of citizens.
"We'll be very true to the mandate which we sought from the people."
Read more: Mary Lou McDonald - 'I may well be the next taoiseach'
