Dutch driver Verstappen, who has won a record 14 races in a single season, told reporters there had been "a constant kind of digging, being disrespectful" and especially from one individual.
"At one point it's enough. I don't accept it...you can't live in the past, you just have to move on," he told reporters.
There was no immediate comment from Sky, who also have access to interviews carried out by Formula One television.
Verstappen won his first championship last year at the Abu Dhabi season-ender, with the Red Bull driver going into the race level on points with Mercedes Lewis Hamilton.
Then-race director Michael Masi's controversial change to the safety car procedure gave Verstappen the chance to pass his Mercedes rival on the last lap and deny him a record eighth title.
Some Hamilton fans feel the Briton was robbed and have been vocal on social media, where Verstappen's 'Orange Army' is similarly active.
Verstappen suggested some of the commentary had helped stoke hostility.
"You keep disrespecting me and at one point I'm not tolerating it any more," he said.
Red Bull principal Christian Horner said he took the decision to shun Sky after comments at the US Grand Prix where Red Bull won the constructors' title for the first time since 2013.
In Mexico, the governing FIA announced the team had accepted a $7 million fine, and reduced wind tunnel time for 12 months, as the only team to have breached last year's cost cap by spending more than allowed.
"Next race, service will be resumed. we just wanted to lay down a marker to say certain things aren't acceptable and as a team we stand together," said Horner of his team's boycott of the broadcaster's English, German and Italian channels.
Sky has the rights to live coverage in the UK and Ireland and also provides coverage and commentary for Walt Disney's ESPN networks in the United States.
"Accusations of championships being robbed is something we don't feel is an impartial commentary...Max is very upset about it and as a team we support him fully," said Horner.
