Dr Yudhoyono tweeted a photo of himself reading the letter.
A spokesperson for the President said the contents of the letter was "in accordance to our expectations".
Dr Yudhoyono says bilateral co-operation will remain suspended until the Australian government signs up to a code of ethics.
"We must take precise measures, so that what was happened previously won't be repeated and the future steps must start with mutual trust between Indonesia and Australia.”
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Meanwhile, Indonesian Trade Minister Gita Wirjawan, who last week warned the spy drama could prompt Indonesia to seek alternative trading partners, has confirmed the cattle trade in particular was under review.
Mr Gita last week asked the Indonesian parliament to examine changing animal health to allow the import of live cattle from countries other than Australia, including those with foot and mouth disease such as Brazil.
Asked on Tuesday whether the fallout over the spying controversy had affected those plans, Mr Gita said: "I think you can attribute some of that or a lot of that to the unfortunate incident that took place.
"Yes, we are reviewing the bilateral posture and we are looking at alternatives to the optionality that we have on the table.
"There are other places that I think can help us with our food security aspirations. We are looking at those."
However, shipping live cattle to Indonesia from Australia is much cheaper than doing so from other countries. The cost of exporting a beast from Darwin to Jakarta is about $100 to $150, on top of the price of the animal, compared with about $800 if it comes from Brazil.
Mr Gita appeared to concede Indonesia, where beef prices remain high as the country struggles to source enough supply, would not be able to completely dislodge itself from Australia in terms of cattle imports.
He said the Indonesian government was "mindful of the need to maintain stability in prices".
Mr Gita said the phone-hacking scandal had led to a lack of trust on the part of Indonesia, but believed relations would normalise.
"We've always considered the Indonesia-Australia relations ... to be way up there in terms of the degree of importance," he said.
"I think the ... expressions that have been made by Indonesians reflect the lack of trust as a result of the unfortunate incident."
"I think there are ways to remedy the relationship."
