"Then, on-the-record with our journalists & editorial columnists (sic)", Murphy tweeted. "Not nice", Trump tweeted.
But on Tuesday, he said he would "keep an open mind" about pulling the United States out of the landmark, multi-national Paris Agreement on climate change - he'd said in the campaign he would yank the US out - and he allowed, "I think there is some connectivity" between human activity and climate changes.
Trump, who has yet to hold the traditional news conference held by a president-elect in the days after winning, said his own businesses are "unimportant to me" in comparison to the presidency, but he also said he now believes he could continue to run them at the same time if he wanted.
As for global warming, Trump has repeatedly questioned the idea, suggesting at times that it is a hoax perpetrated by the Chinese to hurt US manufacturers with environmental regulations.
His meeting with television executives triggered reports that he lectured them about their coverage of his campaign.
Early on Tuesday, the USA president-elect tweeted that he had called off his meeting with what he always refers to as the "failing" New York Times. "We did not change the ground rules at all and made no attempt to, she added in the statement "They tried to yesterday - asking for only a private meeting and no on-the-record segment, which we refused to agree to".
The top spokeswoman for the newspaper, Eileen Murphy, contradicted Trump's account in an email to NPR. There will also be an off-the-record session with the newspaper's publisher.
The apparent U-turn came a day after he berated media chiefs at Trump Tower for their "unfair" election coverage.
Canceling meetings at the newspaper could have allowed him the chance to avoid direct questioning on the subject.
Trump tweeted early Tuesday that he backed out "when the terms and conditions of the meeting were changed at the last moment". Instead, Trump took to Twitter to attack the paper, describing it as "failing" and "nasty", and accusing the paper of changing the terms of the sit-down at "the last minute". I can fully understand that - but why announce?
He said the transition team "is working very smoothly, efficiently and effectively" and meeting with "patriots" "many of whom will soon be a part of our government". The president-elect frequently refers to the paper as "failing".
Gallup has asked the unity question after the previous three presidential elections, and found "Americans are less optimistic about Trump bringing the country together than they were about Bush and Obama in those instances".
Those in attendance included "Meet the Press" anchor Chuck Todd; ABC correspondent Martha Raddatz; anchors George Stephanopoulos and David Muir of ABC News; Wolf Blitzer and Erin Burnett of CNN; "NBC Nightly News" anchor Lester Holt; and a group of CBS anchors including Norah O'Donnell, Charlie Rose, John Dickerson, and Gayle King.
Kellyanne Conway said on MSNBC on Tuesday that Trump is setting a tone for congressional Republicans by refraining from calling for more investigations.