Documents revealed yesterday by the ABC and Guardian Australia show that Australia attempted to intercept president Susilo Bambang Yudhoyono's phone on at least one occasion and monitored his calls for 15 days in 2009.
Mr Abbott refused to be drawn on the allegations, but Jakarta has responded publicly, withdrawing its ambassador and demanding Canberra come clean on the electronic surveillance program.
In his first public comments on the scandal, Mr Yudhoyono voiced his displeasure at the latest allegations, and singled out comments made yesterday by Mr Abbott in which he said "all governments gather information".
"The actions of US and Australia has very much wounded the strategic partnership with Indonesia, a fellow democratic state," he tweeted.
"I also regret the statement by the Australian PM that belittles this surveillance to Indonesia, as if no wrong has been done."
Australia has been given two days to respond to Indonesia over revelations it spied on the president, as Indonesia puts asylum seeker cooperation under review.
As well as Mr Yudhoyono, other spying targets included the phones of first lady Ani Yudhoyono and key government figures.
Before leaving for Jakarta, ambassador Nadjib Riphat Kesoema told media at Canberra airport that he did not know how long he would be kept out of the country.
"I think a good explanation will be the best way to ease the problem," he said.
Indonesia's coordinating minister for politics, law and security, Djoko Suyanto, earlier said he wanted an explanation from Mr Abbott within two days.
Mr Suyanto is the man Mr Yudhoyono has put in charge of dealing with asylum seekers and people smuggling, including whether or not to help Australia tackle the problem.
He appeared on national television last night calling for an explanation.
"The problem is that I haven't received any explanation from the Australian Government relating to the news," he said.
"That's what the Australian Government should explain, that's why it becomes a very important message to be explained by them.
"I can understand the dilemma faced by Tony Abbott because the tapping happened in 2009, not during his administration."
Asylum seeker cooperation 'absolutely' under review
Mr Suyanto says all cooperation with Australia is now under review.
Indonesian foreign minister Marty Natalegawa says that "absolutely" includes asylum seekers.
"There's a message we have to send, that things like this in the 21st century - we are not talking about the Cold War era - in the 21st century, the wiretapping issue should have been far behind us," he said.
"I want to make it absolutely clear. An unfriendly act, unbecoming of a relationship between strategic partners."
Mr Abbott refused to be publicly drawn on Indonesia's call for an explanation when speaking to reporters this morning in Canberra.
"I'm just not going to say anything or do anything that could possibly damage our close friendship and our strong cooperation with Indonesia," he said.
"I'm just not going to say anything about intelligence matters. It's never been the custom or practice of Australian governments to comment on these things, and I don't propose to change that practice now.
"The truth is that we have a very good relationship with Indonesia. Obviously today may not be the best day in that relationship, but nevertheless we do have a very good and strong relationship with Indonesia.
"It's in no-one's interest to do anything or say anything that would jeopardise that relationship and certainly I'm not going to."
Opposition Leader Bill Shorten says he expects the Prime Minister to represent Australia's interests positively and constructively overseas.
"I and Australians expect Prime Minister Abbott to do everything he can to keep our relationship with Indonesia strong," he said.
Greens MP Adam Bandt says the relationship with Indonesia is rapidly deteriorating.
"That's why Tony Abbott needs to very quickly make it clear whether or not this kind of activity is still occurring and whether or not we view the Indonesians as partners and neighbours or whether the Indonesian president and his wife are treated as a national security threat," he said.
Carr blasts US for allowing leaks in the first place
The spying revelations are part of the trove of leaks by former US National Security Agency worker Edward Snowden.
Former foreign minister Bob Carr has attacked the US for allowing the sensitive documents to be made public.
Mr Carr says the United States would be critical of any other nation that failed to prevent the release of such material.
"Certainly if it had gone the other way, if there'd been some official in Canberra, some contractor in Canberra, who allowed a slew of material as sensitive as this to be plastered over the world's media, America would be saying very stern things to someone they'd be regarding as a woefully immature ally and partner," he said.
But Mr Carr has warned that governments should not authorise intelligence gathering simply because it can be done.
"All governments have got to be aware of what might be called the cult of intelligence," he said.
"The appeal that intelligence services probably have to get material that involves an unacceptable risk.
"When you've got the technology, when you've got the capacity to suck up material out of the ether, then it's hard to resist it."
Mr Carr says Australia should apologise to Indonesia.
"I've suggested today that first of all we apologise to the Indonesians, this is going to happen sooner or later, it might as well as happen sooner.
"Second, that we make a commitment to the Indonesians that approximates that made by president Obama to the Germans in respect to revelations from the same source about surveillance of Angela Merkel's phone calls."
But another former foreign minister, Alexander Downer, believes the Government should neither confirm nor deny spying on Indonesia.
He thinks Australia's intelligence capability would be undermined if the Government sought to apologise.
"I think what Tony Abbott should do is just handle the relationship as smoothly and as calmly as he can, and I don't think he should overreact," he said.
"(The spying) didn't happen when he was the Prime Minister, it happened when Kevin Rudd was the prime minister. The Indonesians know that and I think there's nothing Tony Abbott and his team can do but just wait until the storm passes."