Gibson, 29, apologised and is facing an internal disciplinary process.
"Players should be more responsible. Social media plays a massive part," Grayson told BBC Radio 5 live.
"The club has dealt with it internally. He has admitted and apologised that he was bang out of order."
Grayson left Preston for Sunderland in June, with the task of reviving a club relegated from the Premier League last season.
The Black Cats start the 2017-18 Championship season against Derby on Friday.
Jermain Defoe, Jordan Pickford and Fabio Borini have all left the Stadium of Light, while the Black Cats have brought in more modest recruits such as Aiden McGeady, James Vaughan and Jason Steele for the challenge ahead.
"There isn't a great deal of money to spend," Grayson added. "We've got a decent enough budget but it's still way short of a lot of other clubs around us.
"The players we've brought in so far haven't been significant signings compared to the rivals who have come down spending huge amounts of money.
"What I'm trying to do with (chief executive) Martin Bain is get the club some stability, where they don't have to spend millions of pounds, and that they can do it on a lesser budget than Sunderland have been used to.
"We're trying to bring some pride back to the area and the club. Over the past few years it's been in a dark place and mismanaged."
Sunderland manager Simon Grayson says players must be more aware of their public presence after midfielder Darron Gibson was filmed in a video appearing to criticise some of his team-mates.