"We owe it to the families of the people lost on that plane to try to solve the mystery. We owe it to the travelling public, who obviously want to be confident of their safety in the air," Abbott told a local radio station, The Australian newspaper reported.
Last week's discovery of wreckage on the French island of Reunion off the African coast "does seem to confirm that (MH370) went down in the Indian Ocean," Abbott said.
"It does seem very consistent with the search pattern that we've been using for the last few months."
Aviation experts in southern France began examining the wreckage - apart of an aeroplane wing - on Wednesday.
At a press conference in Paris late Wednesday, French prosecutor Serge Mackowiak spoke of a "very strong supposition" that the wreckage part was from the Boeing 777 used for flight MH370.
Malaysian Prime Minister Najib Razak went further, telling media in Kuala Lumpur that experts have "conclusively confirmed" that the wreckage was from MH370.