Speaking during his regular 8pm slot on Thursday evening, Mr Neil admitted it had been a "rocky start".
He told viewers he would be away for a "few weeks" and return "before the summer is out". Colin Brazier is to step in for the former BBC journalist.
The controversial channel, which critics have accused of stoking US-style "culture wars", has had a difficult start since its launch on 13 June.
It received 373 complaints from viewers of Tonight Live with Dan Wootton after the former Sun executive editor delivered a monologue criticising the then expected delay to England's final easing of lockdown restrictions.
Mr Wootton claimed some of the medical experts and politicians tasked with managing public health of wanting to create an "ultra-cautious biosecurity state" copying "the likes of China".
Some critics had described the new channel as a UK version of America's Fox News, claims Mr Neil said were unfair and off the mark.
In its opening days, GB News was also hit by technical issues and a number of brands have announced they will no longer advertise on the channel, citing concerns about its content.
Accepting it was a tough start for the channel, Mr Neil said "we get better every day and there’s clearly an appetite for what we’re doing."
He said: "And in two short weeks, we’ve already built a loyal audience which has beaten all of our expectations, it’s often bigger than the other news channels and it’s growing.
He added: "That’s the real story about GB News to date and you won’t often read that in the papers.
"So on behalf of GB News, I say to all of our viewers, thank you. We won’t let you down and you ain't seen nothing yet."