Mr Cowen had to deny being drunk or hungover during the RTE interview.
The taoiseach was interviewed live from his party's conference, which is taking place in Galway.
On Wednesday he said: "I'm sorry that it emerged in the quality it did. There was a hoarseness in my voice throughout the day yesterday."
He added: "It wasn't my best performance and I would like to apologise for that.
"I would hate to think the reputation of the country or the office of taoiseach would in any way be affected by what I had to say."
Mr Cowen again denied any suggestions he was hungover.
"The assertions made subsequent to it were without justification, without foundation, were not correct, were not true," he said.
Mr Cowen had attended the Fianna Fail party dinner at the Ardilaun Hotel in Galway on Monday night.
Following the interview, the finance spokesman for the Republic's opposition claimed there were concerns over the leadership of the country.
"There was a feeling this morning that this can't continue, the game is up," said Fine Gael's Michael Noonan.
Simon Coveney, also of Fine Gael, who said in a Twitter message on Tuesday that Mr Cowen sounded "half-way between drunk and hungover" in the interview, has said he accepted the taoiseach's apology.
Mr Coveney said that he believed Mr Cowen, but added that he did not regret his previous tweet.
He said it took "a big person to apologise when big mistakes are made as was the case yesterday morning and the Taosieach should be recognised for that."

Charles Raine, 66, known as Neville, and his nephew Philip Raine, 47, were sentenced to three years in prison.
The pair, from County Durham, were last month found guilty by a jury at Teesside Crown Court of conspiracy to use criminal property.
Police investigating missing ewes organised identity parades to reunite them with their rightful owners.
The court was told the value of the sheep was just under Â£25,000.
Phillip Raine's partner Shirley Straughan, 41, was cleared of the same charge.
In all, 16 farmers lost sheep between 2010 and 2013.
In victim impact statements read out in court some said they felt "betrayed".
More than 115 animals were identified as coming from farms in County Durham, North Yorkshire and Cumbria, despite usual markers being removed.
Identity tags were found on the Raine's farms in Bowes.
Speaking after the sentencing, Det Insp Paul Phillips of Durham Police, who led the investigation, said: "This inquiry has revealed the complete contempt the two men convicted today have for other farmers.
"They have broken the rules of the unwritten 'Shepherd's Bible' which has guided sheep farmers for generations. Farmers need to be able to trust their neighbours and they have shown themselves unworthy of that trust.
"We believe the force is the first to secure a prosecution under the Proceeds of Crime Act connected to the theft of sheep.
"Hopefully, this will... send a strong message to those who might be tempted to steal their neighbour's animals that if they do, will be brought to justice."

The future of the outdoor pool in Milford Haven was uncertain after the county council voted to cut its Â£1,000 annual budget.
But the town's Astoria nightclub and the Church at Nantucket donated Â£1,000 to keep it open this year.
Members of the Milford Haven Ladies Circle also offered to prepare the pool ready for its opening in July.
Pembrokeshire council said it would work with elected members, the town council and the community to "ensure a bright future for the town's paddling pool".
Mayor of Milford Haven, Colin Sharp, said the pool had "always been an important feature in Milford Haven" but its future was still unclear.
Irish Prime Minister Brian Cowen has admitted that a controversial radio interview he gave on Tuesday was not his "best performance".
[NEXT_CONCEPT]
Two farmers convicted of possessing more than 100 sheep which did not belong to them have been jailed.
[NEXT_CONCEPT]
A Pembrokeshire paddling pool which had its funding cut will open this summer thanks to donations.