The Easter Road striker, 22, allegedly told a man to go "back to your own country ya clown" during an online exchange.
The Scotland international, who is on loan at Hibs from npower Championship side Wolves, allegedly made the comment while responding to a tweet from a Twitter user believed to be based in Livingston, West Lothian.
His Wolves deal expires at the end of the season but the club have an option to extend it until the summer of 2014.
Wolves paid Dundee £150,000 to sign Griffiths in January 2011 but his only appearance for the club came in the League Cup tie against Northampton the following August, before he was farmed out to Hibs for his first loan spell at Easter Road last season.
A spokesman for Lothian and Borders Police said a 22-year-old man had been arrested and charged under the Communications Act in connection with an incident on January 10. The spokesman added: "He has been bailed to appear in court at a later date."
A spokesman for Hibernian declined to comment.
Increasing numbers of Communications Act 2003 convictions have prompted debate of its modern usage.
It was earlier mooted that Twitter account holders with few followers could escape prosecution.