While saying he "regretted" the diplomatic row, Abbot said he didn't think Australia had a reason to apologize, the Australian Broadcasting Corp. reported Tuesday.
An Indonesian presidential spokesman said Susilo Bambang Yudhoyono "regretted" Abbott's response.
Relations between the two countries degenerated Monday when Indonesia recalled its ambassador in Canberra after documents revealed by the ABC and The Guardian-Australia indicated Australia tried to intercept Yudhoyono's phone on at least one occasion, and monitored his calls for 15 days in 2009.
On Tuesday, Yudhoyono posted on his Twitter page criticism of Abbott for earlier comments in which he had said that "all governments gather information."
"The actions of U.S. and Australia have very much wounded the strategic partnership with Indonesia, a fellow democratic state," Yudhoyono tweeted. "I also regret the statement by the Australian PM [prime minister] that belittles this surveillance to Indonesia, as if no wrong has been done."
Speaking before Parliament, Abbott said Australia's security was paramount, ABC said.
Abbott also repeated his assertion that "every government gathers information," adding that every government "knows that every other government gathers information."
He also told Parliament that Australia shouldn't be apologetic for "steps we take to protect our country."
"Importantly, in Australia's case, we use all our resources, including information to help our friends and allies not to harm them," Abbott said.