"You have the second-highest carbon emission per person on earth and you're burying your head in the sand. You're not a climate sceptic, you're a climate denier," Ms Tobin said.
After the interview, the Sutherland Shire MP posted on Facebook "oh no! Ignorant Pommy weather girl calls me a climate change denier".
The post has since been deleted.
But Ms Tobin was quick to respond, correcting Mr Kelly and listing her educational and professional experience.
"I'm a meteorologist: a degree in physics and meteorology, four years as an aviation forecaster at the RAF, twelve years as a broadcast meteorologist, attended a WMO climate course last year and up to date with all the science," she tweeted, with a hashtag #NotAWeatherGirl.
Morgan also took to Twitter after the interview to continue criticising Mr Kelly.
"Here are some facts, Mr Kelly: 1) Australia just had its hottest year on record. 2) Most scientists say climate change is driving the heat higher. 3) Australia's currently suffering one of its worst-ever national bushfire seasons. 4) Do the maths," the host said.
Earlier on the show, Mr Kelly insisted, "I follow the science".
"And if you look at our science, if you look at the long-term record rainfalls in Australia there is simply no trend," he said.
"To try and make out - as some politicians have - to hijack this debate and exploit this tragedy, to push their ideological barrow that somehow or other the Australian government could have done something to reduce its climate emissions that would have reduced these bushfires is just complete nonsense."
The MP has since apologised.
"I apologise to her. There was a bit of quid pro quo to it, it was at the end of a rather combative interview," he told The Sydney Morning Herald.
He said he used the word "pommy" as a term of endearment, but conceded "weather girl" was probably a step too far.
"But she used the 'denier' slur and basically cut me off and didn't give me a chance to respond," he said.
A former regular guest on Sky News Australia, Mr Kelly was in September last year denied permission by the Prime Minister's Office to appear on the ABC's Q&A program.
That followed his comments at an Australian Monarchists League function where he claimed the island nation of Tuvalu was "floating, not sinking [due to climate change]" because the island is a coral atoll and "a coral atoll actually floats on the ocean".
Since the first fires began in Australia in August, more than five million hectares of land has been scorched.
Twenty-four people have been killed in the fires, an estimated 500 million animals are dead and more than 1,600 homes destroyed.