Indonesia demanded that Australia apologise over claims that Australian spies in 2009 targeted the mobile phone of Indonesia's President Susilo Bambang Yudhoyono, along with those of his wife and closest confidants, after the Australian Broadcasting Cooperation's (ABC) report on Monday revealed details of the spying activities, Xinhua reported.
Indonesia recalled its ambassador from Canberra on Monday and is reviewing all cooperation with the Australian government.
According to media reports, the Indonesian president has lashed out at Tony Abbott in a series of angry tweets, accusing him of taking the spying scandal too lightly.
In a statement to the Australian parliament on Tuesday, Abbott said Australia should not be expected to apologise for the steps it took to protect itself now or in the past, "any more than other governments should be expected to apologise for similar steps that they have taken".
"The first duty of every government is to protect the country and to advance its national interests," Abott said.
Abott said he wanted to make it "absolutely crystal clear" that Australia has "deep respect" for Indonesia.
"I regard President Yudhoyono as a good friend of Australia, indeed as one of the very best friends that we have anywhere in the world," he said in the House of Representatives.
"That's why, Madam Speaker, I sincerely regret any embarrassment that recent media reports have caused him."
"It is in everyone's interests - Indonesia's no less than Australia's - that cool heads prevail and that our relationship grows closer, not more distant," Abbott added.
Meanwhile, the head of ABC has defended its reporting of the Australia-Indonesia spy furore, saying the story is in public interest.
"I think it was an important story that should have been told," ABC Managing Director Mark Scott told a Senate estimates hearing in Canberra on Tuesday.
According to him, while the story might be embarrassing and could cause some short-term difficulties between the two countries, a debate was taking place all over the world on the reach of government spy agencies in the digital age.
"What information can be procured, what information can be shared - I think the story yesterday (Monday) centrally went to that," Mark Scott said.
"It caused some damage to Australia in the short term, but I think we would say that reporting was absolutely in the public interest and it was probably in the national interest in the long term," he added.