In a letter to the new conservative leader, the group said Australia could make a significant difference in promoting rights in the region with the "right mix of pressure and engagement".
"Pre-election you have stated that your party's foreign policy will be 'designed to protect and project our reputation as a strong and prosperous nation and our values as an open liberal democracy'," the letter to Abbott said.
"Promotion of these values should include publicly raising human rights concerns with foreign leaders," it added.
HRW said it was "concerned" Abbott failed to raise human rights with Indonesian, Chinese and Vietnamese leaders in recent meetings on the sidelines of several Asian summits including the Asia-Pacific Economic Cooperation meeting.
"We encourage Australia to play a leading role in the region, calling for and supporting transitions to open liberal democracy," it said.
Abbott, who made Indonesia his first overseas stop as prime minister, has said that when it comes to foreign policy, Australia has to have "a Jakarta, not a Geneva focus" but defended his decision not to raise human rights concerns in Brunei.
"We will say our piece when there are major human rights abuses taking place but, generally speaking, it's not the job of the Australian prime minister to stand up and give lectures to the wider world," he said last week.
Human Rights Watch focused on 15 countries it operates in in its letter, including Australia's key trading partners China and India and close regional neighbour Indonesia.
It said human rights concerns in the group -- which included Afghanistan, Bangladesh, Burma, Cambodia, Fiji, Malaysia, Nepal, Papua New Guinea, the Philippines, Sri Lanka, Thailand, and Vietnam -- involved crackdowns on freedom of expression, assembly, and association; repression of religious minorities; and failure to hold security forces to account for torture, killings and disappearances.
"Australia should recognize that a secure Asia-Pacific region depends on countries working together to address human rights problems," said HRW Australia director Elaine Pearson, who signed the letter along with executive director Kenneth Roth.
"Prime Minister Abbott shouldn't give other countries in the region a free pass on human rights, just as he shouldn't neglect important rights issues at home."
On domestic issues, Human Rights Watch called on the Australian government to address several issues, particularly the treatment of asylum-seekers and refugees and same-sex marriage.