Morrison's op-ed follows weeks of debate around the appropriateness of colonial era statues, and Australia Day itself, considering the country's indigenous history.
A statue of Captain Cook in Sydney's Hyde Park that was defaced last week has now been fenced off to protect against vandalism.
"However, our modern Australian story has not been without its failures, its shames and tragedies, in particular for indigenous Australians," Morrison conceded.
Greens MP Adam Bandt said calls to modify plaques on statues of colonial leaders like Captain Cook were part of a broader movement to include indigenous perspectives.
Australia needed to ensure that when it was "commemorating or celebrating people in this country, whether it's by statues or any other way, it's inclusive".
"This is the beginning of a growing debate and push for change in the Australian community so we can celebrate our national day on a day that means something for everyone," Bandt told reporters in Canberra on Monday.
Victorian councils, Yarra and Darebin, had their right to hold citizenship ceremonies removed by the federal government after voting to ditch Australia Day celebrations on January 26.
Nationals MP Andrew Broad said it was important people were educated about Australia's past, as well as its indigenous history, but the statues should be left as they are.
"I don't think we do well to rewrite history," Broad said.
"Statues that were put up in their time can be an observation of what the thinking was at the time."
Opposition frontbencher Tony Burke criticised the government for wading into the debate over Australia Day and indigenous history in the last two non-sitting weeks.
"A prime minister who's focused at throwing mud at his political opponents, talking about statues and talking about councils in Melbourne - I think says it all about this prime minister," Mr Burke said.
