Prime Minister Jacinda Ardern, who initially defended Jones' comments as expressing his opinion, said that calling for Carter's head to roll was a step too far and reprimanded Jones.
She did not comment on whether Jones should attend the Obama event, saying it was a matter for him.
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."
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."
He 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."
The Government owns a 52 per cent share of Air NZ and has a say in the composition of the board, which is up for renewal in at the AGM in September.
But it does not have a say in how the board runs the airline's commercial operations.
Luxon defended Air NZ's record in the regions, saying it 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."
