Claims that Australian spies targeted the phones of Yudhoyono, his wife and ministers in 2009 has sparked a snowballing diplomatic crisis.
Hundreds of protesters in Jakarta pelted the Australian embassy with eggs Friday, while Indonesia has frozen cooperation on people-smuggling and joint military exercises and recalled its ambassador from Canberra over the scandal.
Yudhoyono has written to Abbott, who he has publicly lambasted on Twitter for what he called a lack of remorse, calling on him to fully explain the allegations.
Abbott, who has expressed regret at Yudhoyono's embarrassment but refused to apologise for intelligence gathering, said Saturday he had replied to the letter but did not discuss its detail.
"The letter is now in the process of being delivered," he said.
"I think it would be wrong of me to talk about what I've said to the president before the president has actually received it."
Abbott said Canberra was committed to a strong relationship with Indonesia, a vital strategic and trading partner.
"There'll be good days and there'll be better days," he said.
"But my determination is to ensure that the relationship is constantly improving and my gratitude is always to President Yudhoyono, who has been a very good friend to Australia -- one of the best friends we've ever had."
Indonesia has reacted furiously to the spying reports, which were based on leaked documents from US intelligence fugitive Edward Snowden.
They showed that Australia's electronic intelligence agency tracked Yudhoyono's activity on his mobile phone for 15 days in August 2009, when Labor's Kevin Rudd was prime minister.
At least one phone call was reportedly intercepted.
The list of tracking targets also included Yudhoyono's wife Ani, Vice President Boediono, former vice president Jusuf Kalla, the foreign affairs spokesman, the security minister and the information minister, the reports said.