Communications Minister Mitch Fifield says his department secretary will hold an inquiry into the matter and report as soon as possible.
The ABC's board selects the managing director, while the governor general of Australia appoints the board and chairman at the recommendation of the government.
"Mr Milne has no understanding of editorial independence, proper complaints handling processes, or the appropriate distance a board chair needs to keep from staffing matters", the union said in a statement.
One of the critical developments earlier this week was the revelation that Milne had pressured Guthrie to fire chief economics correspondent Emma Alberici, whose reporting had been under fire from the office of then-Prime Minister Malcolm Turnbull. I just think it's simple.
The ABC, which is the equivalent of the BBC, is the nation's most trusted news source, according to public surveys, but regularly runs into trouble with the government of the day over its political coverage.
"I feel that the interests of the ABC have always been uppermost in my mind".
Prime Minister Scott Morrison welcomed Milne's exit on Thursday, saying it was time for the ABC to "resume normal transmission".
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Acting Associate Attorney General Jesse Panuccio, a former general counsel for Florida Gov. They say the president should not fire him unless he believes he is lying.
"They [the government] hate her", Milne reportedly wrote. Get rid of her.
Milne said then the board's decision was made in the "long-term interests of our own people and the millions of Australians who engage with ABC content every week". "There is no guarantee they (the coalition) will lose the next election".
Guthrie was abruptly sacked on Monday and has expressed she is looking into her legal options. "Nobody (from the government) ever told me to hire anybody, fire anybody, or do anything else", he told the ABC's current affairs program 7.30.
Separately, another publisher reports that Milne had also ordered the firing of ABC's political editor, Andrew Probyn, due to government criticism.
Mr Milne, who resigned on Thursday, is also the chairman of two ASX-listed companies - accounting software giant MYOB and communications technology business NetComm Wireless.
Turnbull and his supporters have openly claimed that media mogul Rupert Murdoch said that "Malcolm had to go" in a comment which allegedly spurred an editorial campaign to remove him across News Corporation's mastheads.
On Thursday, Milne, a former executive at Australian telecom giant Telstra, described the recent reports as a "firestorm" and said he chose to quit because he "wanted to provide a release valve" for the network.