Shaw resigned over a post he made regarding the deputy prime minister, and unreservedly apologised for comments made about Peters on LinkedIn.
"He's not truthful. He's not accurate. He's malicious and he is here on behalf of international tobacco," Shaw wrote. "His return is the worst of this gang of thugs."
Shaw was responding to comments from Peters on Monday, during which he accused state-funded media organisations of a lack of independence from the previous government.
Hipkins told Morning Report the board member's comments were not appropriate in the context of his role. NZ On Air administered the funding.
"As a member of the NZ On Air board, I think it's very important that Andrew Shaw upheld sort of the independence of NZ On Air, the political independence of New Zealand on air as an entity. So I think he's done the right thing in standing down."
But he said Peters' recent comments about media independence and journalism funding were outrageous, considering he was part of the coalition government that established the PJIF.
"In fact, he was the deputy prime minister in it. [The PJIF] was established during the period of, you know, Covid-19 instability and uncertainty to ensure that the media could continue to do a very, very important job, which was to scrutinise the actions of government - and at all levels, including local government - at a very extraordinary time for the country."
He said Peters' accusation of "bribery" was "totally baseless".
"The person who needs to do something about it is Christopher Luxon - this is his deputy Prime Minister making very serious allegations without any basis and without any facts to back them up.
"You know, ultimately Christopher Luxon is in charge, he is responsible for ensuring that ministers uphold the standards of being ministers. Winston Peters is not and I think Christopher Luxon should do something about that…
"He should certainly be pulling Winston Peters into line, although at this point, it seems like Winston Peters is running the show rather than Christopher."
Hipkins said he had "absolutely no idea" what specifically Peters was referring to when he said journalists should "tell the public what you signed up to, to get the money".
"I've long since given up trying to second-guess what Winston Peters is saying."
He noted former broadcasting minister Willie Jackson in an interview this week said it was NZ On Air that established the criteria for funding, not the government.
Hipkins, who was prime minister up until Monday, said he saw no evidence the fund "compromised" newsrooms' ability to scrutinise the government.
Luxon on Tuesday told Newshub he had not seen Peters' comments, despite the widespread news coverage.
Elsewhere in the interview with Morning Report, Hipkins called the government's decision to help fund its tax cuts by cutting smokefree efforts as "morally reprehensible", and said Labour's new shadow Cabinet would be announced later this week.