Although Taoiseach Enda Kenny has reappointed James Reilly as Fine Gael deputy leader, Mr Martin said there is no such position in his party.
Mr Martin said that his aim, now, was for the party to gain a seat in all constituencies where they had none, and a second in other constituencies.
Fianna Fáil have not appointed a new deputy leader since the front bench reshuffle of July, 2012. Éamon Ó Cuív, Brian Cowen, Bertie Ahern, Ray McSharry and Mary O’Rourke have all been deputy leader in the past.
In an interview with the Irish Examiner, Mr Martin said: “There is no position, it just doesn’t arise.
“It hasn’t been on for a long time.” On the issue of Brexit, Mr Martin said it is now up to all parties — and not just those in government — to push for and protect Ireland’s interests, in the context of negotiations.
“All of us in Europe should use all of our contacts, in the grouping that we are a part of, to articulate Ireland’s key interests, the common travel area, trade and the all-island single market,” he said.
The Cork South-Central TD praised Mr Kenny for “endeavouring to make sure that the other leaders get it that Ireland is a special case, because of its uniquely historic relationship with the United Kingdom”.
Mr martin said that he supported Mr Kenny in raising, at the EU council just days after the Brexit vote, Scotland’s unique place in negotiations.
“There is no issue with raising Scotland, there is no big deal about that. I raised it myself in the Dáil. It’s a fair point. A lot will depend on how Britain approaches the negotiations and, ultimately, the optimal outcome for us is that Britain signs up to some sort of Norwegian deal, where Britain still remains part of the single market.
“If Britain has access to the single market, then the issue of tariffs doesn’t really arise in relation to Ireland and Britain.
“That will determine the strength and health of our small- to medium-size enterprises, food companies in particular,” Mr Martin said.
Despite poor poll ratings prior to the general election, Mr Martin said that he was confident that Fianna Fáil would do much better than expected and he now hopes to build on this success.
“My target, all along, was 40 seats-plus. Privately, I would have felt it was always possible to get to the 44.
“I had a lot of self-belief throughout the four years, but the national polls would have continued to sap morale.
“We knew, from our own work on the ground and the calibre of candidates, that we would surprise people.
“We still have a lot of work to do. In the end of the day, we got 25% of the vote. It’s not huge, so I think there is room for growth.
“But I am acutely aware that there are many twists and turns and we are in a very uncertain world,” Mr Martin said.