Marciano spoke before attending a meeting called by Yudhoyono to discuss the issue with Foreign Minister Marty Natalegawa and Indonesia's recalled ambassador to Australia.
Natalegawa said that Indonesia was reviewing bilateral cooperation on issues with its neighbor.
"We have downgraded the level of relations between Indonesia and Australia," he said. "Like a faucet, it is turned down one by one."
Yudhoyono told a news conference after the meeting that he expected a formal explanation if Australia wants to maintain good bilateral relations.
"Clearly, I asked for temporary termination of cooperation on intelligence exchanges and information sharing," he said.
"I also asked for the termination of joint exercises between Indonesia and Australia, either for army, navy, air force or a combination," he said, adding that the snooping reminded him of the Cold War era.
The termination affects cooperation on the thorny issue of people smuggling between the two countries. Indonesia is a transit country for thousands of asylum seekers hoping to reach Australia's Christmas Island by boat. Many people have died while attempting the dangerous journey, and the immigration issue remains a political hot potato in Australia.
Abbott won elections in September on a promise to stop the asylum seeker boats and is relying on Indonesia's cooperation to achieve this goal. He has also ruled out an apology or explanation on the spying allegations.
On Tuesday, Yudhoyono criticized Abbott for not expressing regret over the spying, which reportedly took place in 2009 under a previous Australian government.
In the Australian capital of Canberra on Wednesday, Abbott told Parliament that while he would try to repair relations with Indonesia, he did not "propose to overreact now" to anger over the issue.
"I deeply and sincerely regret the embarrassment that media reports have caused President Yudhoyono, who is a very good friend of Australia, perhaps one of the very best friends that Australia has anywhere in the world," Abbott said. "I do understand how personally hurtful these allegations have been, these reports have been, for him and his family."
"My intention, notwithstanding the difficulties of these days, is to do everything I reasonably can to help to build and strengthen the relationship with Indonesia, which is so important to both our countries," he said.
But Abbott failed to directly answer a question asked by opposition leader Bill Shorten: What progress had been made to restore Australia's relationship with Indonesia?
Abbott, however, said he would respond quickly and fully to a letter Yudhoyono told reporters he was writing to the prime minister.
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.
In Washington, where the U.S. and Australian foreign and defense ministers met Wednesday, U.S. Secretary of State John Kerry said he doesn't discuss intelligence matters in public.
Kerry said the U.S. has an "unbreakable and critical working relationship" with Australia, including in counterterrorism. He said the U.S. also has "great respect and affection for Indonesia" and works with that nation on many issues.
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McGuirk reported from Canberra, Australia. Associated Press writer Matthew Pennington in Washington contribute to this report.
