Ardern said it was "entirely a decision for Shane", adding that she was not going to fire him.
"This is not a sacking offence. He's expressed an opinion, one that I know some New Zealanders will certainly share some sympathy for ... but suggesting someone should be sacked is too far."
Jones said he would not be muzzled by Air New Zealand.
"If anyone on that board believes they are going to muzzle me as a champion for the provinces, then they are sadly mistaken."
Jones denied he was bullying the board, suggesting that their directors fees should be enough of a buffer.
"Check the salaries or the directors fees of the board and if one politician using some florid rhetoric, they conceive that to be bullying, then really.
"They're handsomely paid. They should be able to tolerate political opinion, political challenges ... I accept that Tony will take not an ounce of notice of what I say."
This morning, Jones continued his attack on Air NZ and set his sights on Carter.
"Obviously you'd start with the chairman … I'm telling that board, in terms of the growth and connectivity in provincial New Zealand, it will not increase unless that board changes," Jones told Radio NZ.
Jones also warned Air NZ chief executive Christopher Luxon not to "poke your nose into the political boxing ring, unless you're going to resign today and join the ranks of the National Party".
This afternoon he said his morning remarks had gone through a "process of refinement".
"Both Grant [Robertson] and the Prime Minister have said 'Shane, you have a strong view as a regional provincial champion, but you have no authority to effect changes at the level of the board'.
"And I accept that."
Bridges said Jones should not be accepting corporate hospitality from Air NZ to attend the Barack Obama event, if he was so opposed to their actions.
Jones said he was invited to the dinner, and "as a proxy for the 52 per cent owner of Air New Zealand, I'll go where I like".
"I've got every right to go and listen to one of the greatest western leaders of my lifetime."
Bridges said National had raised concerns about Air NZ's regional services, "but it's how you deal with them and what you've got here is bullying, personal attacks, the calling for resignations of the board, the chair and the chief executive".
"If business wasn't worried, they will be now. And don't think this stuff doesn't have an effect on how they behave, on business confidence and ultimately jobs and employment."
Robertson said today he disagreed with Jones and the board and chief executive were doing a good job.
The Government would consider the board's composition through the normal processes, but Robertson added that the Government was committed to improving regional connectivity through building infrastructure such as roads and rail services.
The Government owns 51 per cent of the company and has a say in the composition of the board.
Air NZ will elect new members of the board at the annual meeting in September, when a third of the seven-person board must resign, but may stand for re-election.
Shareholders only influence commercial operations by electing the board, which has operational independence from shareholders and must adhere to the rules of the NZX.
New Zealand First leader Winston Peters said it was entirely appropriate for Jones attend the Obama dinner.
"He was invited, and I can't go because of having to be here, so I'm pleased he is going."
He said it was the Prime Minister's prerogative to reprimand ministers but refused to back that up himself.
Asked if he had told Jones off, he said "we do not go in for copycat, imitative behaviour. We actually think for ourselves in NZ First."
On Jones' comments that the board was a colony of the National Party, Peters said he had looked at appointments to boards of SOEs and other bodies "and that is the kind of conclusion one might be compelled to have a look at".
"I'll go no further than that, but I've looked at the SOE appointments all over the place and a whole lot of other appointments. And I can tell you if you got a toilet roll out and wrote all the names down you'd use the whole toilet roll up."
He said the board was doing a "pretty good job. All I'm saying is they could do better".
He said the price difference to fly to a regional centre compared to between Wellington and Auckland was "massively different".
"It's cheaper to fly to Australia sometimes."
Luxon returned fire at Jones this morning, saying the airline delivered almost two million more domestic seats and reduced regional airfares 8 per cent over the last three or four years.
"We've got one of the best regional networks of any country on Earth."
Luxon said the Crown had the same rights as any other shareholder.
"That doesn't mean they can dictate the operations of the company. They can't use their majority shareholder position to make the company make non-commercial decisions.
"Decision making is with the company board and the Treasury expects all those decisions to be commercial."
The Air NZ board
• Tony Carter, chairman
• Jan Dawson, deputy chair
• Rob Jager
• Linda Jenkinson
• Sir John Key
• Jonathan Mason
• Dame Therese Walsh
