AMSA had been liaising with New Caledonia search-and-rescue authorities and the cruise ship, with a call going out for nearby vessels to help search.
"As a result, we have now made the extremely hard decision to continue our journey towards Brisbane".
"Our thoughts and prayers are with the guest's family at this distressing time".
A statement from P&O Cruises Australia said the hard decision to suspend the search was made only after expert advice that survival was not considered possible after so long in rough seas.
"A crew member notified the bridge straight away and the "man overboard" incident response was activated immediately", Carnival Australia spokesman David Jones said in a statement. The ship is due back at 11am. After hours of searching in rough seas and high winds, the captain told passengers that "with a heavy heart", they had "made the extremely hard decision to continue our journey towards Brisbane", reports the Courier Mail.
"There is nothing to suggest anything of this kind", P&O said in a statement.
The passenger said suggestions the woman was knocked overboard by a wave were incorrect although she was unsure how the incident occurred on Thursday afternoon near New Caledonia. Police say there do not appear to be any suspicious circumstances at this time, but investigators will meet the ship when it arrives in Brisbane early Sunday.
The ship turned around immediately to search for the woman but Australian maritime authorities chose to end the search at around 07:30 Friday.
A ship passenger, who asked not to be named, described the atmosphere onboard as "very sad, solemn and quiet".
Another woman on board posted to social media that she was "Feeling helpless" and "this woman who fell overboard has a husband and children on board".