A senior Indonesian naval officer, Major Andy Aprivanto, had earlier urged a revision of the policy to turn back boats to avoid casualties in the open sea.
The boat turnaround is part of Abbott's pledge to repeat the success of former Liberal Prime Minister John Howard's draconian measures, although only four vessels were sent back to Indonesia.
His new Government has been talking tough. Bishop said Australia was not asking for Indonesia's permission, but for its understanding.
Howard's Foreign Minister, Alexander Downer, has further stoked the fires by accusing Indonesia of "pious rhetoric".
His comments carry weight in Jakarta because of his former status and his continued influence within the Liberal Party.
"Let me make this point for Mr Natalegawa's benefit - Indonesian-flagged boats with Indonesian crews are breaking our laws bringing people into our territorial waters," Downer told ABC TV's The Drum programme.
"This is a breach of our sovereignty and the Indonesians need to understand that, instead of a lot of pious rhetoric about the Australian Government breaching their sovereignty."
Yesterday, Downer told Fairfax Radio: "There's no point in allowing ourselves to be bullied by the Indonesians in this way. I mean, we have to stand up for ourselves and ... be prepared to call it as it is."
Abbott, who has placed Indonesia at the head of his foreign policy agenda and promised to build a stronger relationship with Jakarta, yesterday tried to play down the gathering storm.
"The last thing I would ever want to do is anything that doesn't show the fullest possible respect for Indonesia's sovereignty," he said.
"The last thing anyone should want is to have Australia's relationship with Indonesia defined by this boats issue, which I am sure will be but a passing irritant."
But Indonesia has made its position clear, abandoning its usually restrained public voice by taking the highly unusual step of releasing a statement detailing the meeting between Natalegawa and Bishop.
The statement quotes Bishop as telling Natalegawa that Australia wanted to co-operate fully with Indonesia "behind the scenes" to avoid bad press that could harm Australia's efforts to combat the flow of asylum seekers.
Natalegawa responded: "It is feared that unilateral steps by Australia may constitute a risk to the close cooperation and trust that has been built within the Bali Process framework and therefore they should be avoided," Natalegawa said.
Yesterday Professor Dewi Fortuna Anwar, a senior presidential adviser, told ABC Radio that "unacceptable" Australian measures risked confrontation.
"If there is any policy that would infringe on our sovereignty - for example Australian Navy making entry into Indonesian waters to chase boats - that would be considered an unfriendly act if not downright hostile that would also be considered unacceptable," she said.
"Let's not be hypothetical about it. Any act by a foreign navy that infringes on a neighbouring country's territorial waters ... could cause incidents at sea."
Interim Labor Leader Chris Bowen said Indonesia's response showed the Government had failed its first foreign policy test - "a very bad omen indeed".
"It is no doubt being watched by the nine other Asean nations for indications of how the Abbott Government will treat the region," he said.
