Australian Foreign Minister Alexander Downer has explained the decision to his Indonesian counterpart Hassan Wirajuda.
The Australian government decided to abandon its legislation rather than face an embarrassing defeat in the Senate.
Jakarta is saying it deeply regrets the failure of the legislation and, in reference to Papua, that it expects Australia to continue to respect the territorial integrity of Indonesia.
But elements within the Indonesian government have hinted it could prompt Indonesia to relax its boat people policies.
An influential Indonesian politician has also urged the Australian government not to give up on plans to toughen its migration laws.
Theo Sumbawanga, chairman of the Golkar party and the Indonesian parliament's foreign affairs committee, says Prime Minister Howard should try again to toughen migration laws.
"Our government should approach the prime minister and the cabinet of Australia to try not to give up in their efforts. Try again," Mr Sumbawanga told ABC Radio.
The government faced one of its most embarrassing political backdowns on the same day it emerged that eight new boat people had arrived on Australian shores.
They are thought to have been dumped by people smugglers on Ashmore Reef in the Timor Sea off Western Australia's far north coast. Defence sources say they are from Burma.
All eight are heading to Christmas Island for health checks but will ultimately be transferred to Nauru for processing.