The Australian Foreign Department website earlier advised Australians not to visit the embassy because of expected protests.
Abbott told Parliament he had received a letter from Yudhoyono demanding an explanation of Australia's phone tapping activity in 2009.
"I want to assure the House that the government will respond swiftly, fully and courteously to the president's letter," Abbott told the House of Representatives.
"As always, my intention is to do everything I reasonably can to strengthen the relationship which is so important to both our countries," he added.
Abbott did not explain how he would satisfy the demand after earlier saying he would not apologize or explain Australia's spying activities.
Abbott also condemned as "tacky" an inflammatory tweet by Mark Textor, a long-term political strategist employed by the ruling Liberal Party. "Apology demanded from Australia by a bloke who looks like a 1970's Pilipino porn star and has ethics to match," Textor tweeted on Wednesday, and then later deleted.
Abbott did not directly answer a question about whether he would review government contracts with Textor's company. "They were tacky comments and they've been withdrawn and apologized for," Abbott said.
The tweet has been reported in Indonesian media and is likely to further fuel anger over the spying controversy.
Textor later tweeted: "Apologies to my Indonesian friends," ''Twitter is indeed no place for diplomacy" and "I agree. Conduct unbecoming."
He has refused to say to whom the tweet referred. Indonesian Foreign Minister Marty Natalegawa has led Indonesian demands for an apology.
With an election year looming, Yudhoyono's tough reaction may in part be an attempt to divert public attention from his troubled party and stoke nationalism.
Analysts describe the furor as the lowest point in an often volatile Australia-Indonesia relationship since 1999, when Australia led a U.N. military force into the former Indonesian province of East Timor following a bloody independence ballot. At that time, Indonesia ripped up a 4-year-old security treaty with Australia. A new treaty has since been signed.
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Associated Press writer Ali Kotarumalos in Jakarta, Indonesia, contributed to this report.