The heated diplomatic exchange between the Pacific neighbours comes after Australian Prime Minister Scott Morrison said Australia is "open" to the idea of recognizing Jerusalem as Israel's capital and emulating the U.S.by moving its embassy there.
"The Indonesian trade minister has made it very clear on the public record that is not an issue of concern to them", he said.
At stake is the government's single-seat majority in the House of Representatives and Morrison's ability to retain power without doing deals with independent lawmakers.
Indonesia's trade minister, Enggartiasto Lukita, denied Australian media reports on Tuesday that Jakarta was considering putting the pact on hold over the possibility of Canberra changing its stance on Israel.
Since taking over as Prime Minister in August, Mr Scott Morrison has presented himself as a no-nonsense leader focused on the everyday interests of ordinary Australians. West Jerusalem, where the Knesset and the other main organs of the Israeli government are located, has been sovereign Israeli territory since 1948.
"We will continue to strengthen ties between", he said in the Tweet.
Indonesian Foreign Minister Retno Marsudi hit back in response, saying Indonesia would continue to watch Morrison's statement closely on the issue.
Indonesia is home to the world's largest Muslim population, with almost 90% of the country's citizens, around 225 million, adherents of the religion.
"We do not understand how moving the embassy and recognizing Jerusalem as the capital of Israel will preserve the peace process, while a vote at the General Assembly today rejecting the authority of the delegation of the State of Palestine only during its presidency of the group of 77 China will undermine those opportunities?"
But Mr Morrison - who fell into line with the Government's then-position at the time - said he was considering a re-think of Australia's position.
In a May article, Sharma wrote of the USA embassy move: "Now that the United States has broken the taboo, as a pragmatic and solution-oriented nation Australia should be prepared to lend our support to this common-sense proposition".
Morrison told Parliament that he had briefed Jakarta overnight about the announcement.
The free trade deal with Indonesia is set to unlock education opportunities in the Asian country, which has a workforce of more than 130 million people but half of them only have a primary school education.
For decades, a proposed two-state solution has underpinned the worldwide community's efforts to address and achieve peace in the Israeli-Palestinian conflict ― but Morrison appears unconvinced that it's ever going to work. "Is the prime Minister so panicked about Wentworth he is willing to make the most cynically timed foreign policy decision in living memory?"
Turnbull's government had explicitly distanced itself from the decision by Trump to move the U.S. embassy from Tel Aviv to Jerusalem as "unhelpful" to the peace process.