Morgan was nominated for the 'Music Radio Personality of the Year' prize, but missed out on the 'Gold', 'Silver' and 'Bronze' awards.
BBC Radio 2's Chris Evans took home the top prize, while his 6Music colleague Lauren Laverne won the 'Silver' award.
Meanwhile, BBC Radio 1's Fearne Cotton was handed the 'Best Music Programme' award.
A disgruntled Morgan took to Twitter to vent his disappointment, writing: "They ain't gonna let some dude from NYC win this sh*t. Fern, Lauren, Chris. Yeah right, that's cool? Suckers they come a dime a dozen... (sic).
"I am the peoples champ, don't need no award. Y'all know how it be."
Referring directly to Laverne, he said: "Ask me to cover for her and up her ratings. F*ck that sh*t. Kenickie? Please."
On Cotton's win, he added: "Yo, Fern cotton won a gold award, sh*t is fake!"
He later tweeted: "I don't care about the awards y'all, it's the suckers hat get 'em that got me worried. now go live your life. I'll be back in a week's time."
Returning to Lauren Laverne, he continued: "You got no idea what I put Into my shows. She just shows up and plays in twitter. Ain't that BS."
Laverne later responded to Morgan's comments, tweeting him: "Hi Huey, I really don't know why you're being like this but I do programme my shows, every single morning.
"Just because you don't work like that when you cover for me, doesn't mean it's what I do. Please stop being weird."
She then tweeted her own followers: "Hey all, getting friction from someone I've always got on [well with] here. Very weird, not sure why. But just to be clear, I programme the music on my show every morning. I'd always hoped this was obvious from it being quite good, but apparently clarification would help."
Huey Morgan has since apologised for his comments on the social network site, tweeting his followers: "I am sorry for my comments last night. If I offended anyone, I apologise. I will tweet no more. Goodbye and thank you for you reality check."
The Fun Lovin' Criminals star began fronting his own BBC Radio 6Music show in 2008, and stood in for Laverne while she was on maternity leave in 2010.