The Northern Ireland assembly has not sat for nearly 500 days
Arlene Foster has claimed that some Sinn Féin supporters have told her they will vote DUP because of her party's position on abortion.
She said a lot of people were feeling "disenfranchised" by the result of the Eighth Amendment referendum in the Republic.
"I have had emails from Nationalists and Republicans in Northern Ireland not quite believing what is going on and saying they will be voting for the DUP because they believe we are the only party that supports the unborn," she told Sky News.
"There are many people who are shocked in the Republic of Ireland and whilst I completely acknowledge the result that happened last Saturday, that doesn't take away from the fact that there's a substantial minority of people who feel disenfranchised."
The Irish electorate voted by a landslide - 66% to 34% - to lift the Constitutional ban on abortion, paving the way to unrestricted access to termination up to 12 weeks.
The vote here has placed the spotlight on the North, with pro-choice campaigners urging British Prime Minister Theresa May to intervene and change the law in Northern Ireland too.
When asked about the rights of women facing unplanned pregnancy in Northern Ireland, the DUP leader replied: "Abortion is a very emotive subject, a very sensitive subject and therefore it deserves a serious discussion and a serious, mature debate.
"It certainly does not deserve some of the antics that we've seen recently, frankly, and I did find it, I have to say, quite distasteful to see people dancing about on the streets in relation to the referendum results.
"The way to have that debate, looking at the evidence, speaking to people who have gone through those crisis pregnancies, is to have that debate in the devolved administration.
"We were elected to that devolved administration last March. Nearly 500 days later, we are still not doing the job that we were elected to do.
"There is only one party that is stopping us from doing all of that and bizarrely, that is Sinn Féin, the people who say they want change on in relation to abortion and in relation to same-sex marriage. You can't have it both ways," she added.
"I think what we need to do is be able to put forward a rational explanation as to why we hold those particular views I have a right to hold a different view than others hold. That's called tolerance."
Mrs Foster, whose party found itself holding the balance of power at Westminster a year ago this week, says Mrs May understands that same-sex marriage and abortion are devolved issues.