Morgan left several messages on Twitter taking issue withthe gongs given to Radio 1's Fearne Cotton and his 6 Music colleague Lauren Laverne.
He later deleted the remarks and apologised for his comments.
6 Music controller Bob Shennan told Music Week: "It was a bit of nonsense and Huey is the first to know he's acted in a daft way and he's apologised now and we can move on.
"He's been a valuable part of the 6 story and he's a wonderful broadcaster and much loved, but it was just a blip."
The station's management is also believed to have spoken to Morgan about the incident.
Morgan last year told Digital Spy that he never thought the station would be axed.
Watch Huey Morgan discuss his BBC 6 Music show with Digital Spy below: