She says the Affordable Care Act, also known as Obamacare, has more similarities to the Australian health care system than the GOP's plan does.
"But when everything comes together with the inclusion of Phase 2, we will have truly great healthcare!", Trump added in the tweet sent from his home on his private golf club in Bedminster, New Jersey.
White House spokeswoman Sarah Huckabee Sanders cautioned Friday against reading too much into the president's comment.
Trump described the USA health system as failing, and added that "I shouldn't say this to a great gentleman and my friend from Australia because you have better health care than we do".
In a tweet, Sanders said Trump was right when he praised Australia's system of universal health care in a meeting with Prime Minister Turnbull in NY. But Huckabee Sanders shrugged off that decision.
Businessman-turned-politician Trump, within hours of taking the oath of office on January 20, fulfilled one of his most impassioned promises he had made during his presidential campaign and signed an executive order to "ease the burden of Obamacare", which covers some 20 million Americans. "So maybe when we get to the Senate we should start off with looking at the Australian healthcare system". It allows for state flexibility.
Trump said the United States and Australia are linked by "iron bonds" forged by the Battle of the Coral Sea in World War II and "sealed with the blood of our fathers and grandfathers" who lost their lives halting the Japanese advance in the Pacific.
The video also noted that Australians can expect to live longer than Americans, on average. "Also saves country money!" he tweeted.
"We get along great", Trump said.
Labor's shadow minister for health and Medicare Catherine King said the prime minister was praising a bill that will could lead to thousands of Americans losing their healthcare and "will take away the requirement for health insurers to cover people with "pre-existing conditions" - such as diabetes, autism or cancer", Ms King said in a media release.
Asked why he does not spend more weekends working in Washington, the spokeswoman said: "This is the president's first time to go back home to the NY metro area and I think he's trying to save the tax payers' money the best way he can by taking his team and focus and being in New Jersey instead of in NY where it would have caused a much greater disruption and much greater cost to tax payers".
But while Mr Trump's bill was approved in the House, now it must pass through the Senate, where the legislation is believed to have little support.
Mr. Trump's remarks come after the House on Thursday afternoon passed a bill repealing and replacing large chunks of Obamacare.
"We have a very intense engagement with the United States on these matters", Mr Turnbull said.