The row unfolded when beauty queen Elena Raouna, 23, sent 67-year-old Sir Alan Sugar a cheeky message on Twitter over the weekend.
It read: "Evening sugar ;-) can I call you that? Lol."
The multi-millionaire replied: "yes no problem as long as I can call you fatty".
Fans of Elena, who won the title of Miss British Beauty Curve in 2013, immediately leapt to her defence, branding Sugar a 'bully'.
And now the model herself has demanded an apology from the businessman.
She told The Mirror: "The worst thing about it is I was actually a fan and all I wanted was a simple hello.
"I kind of got excited when I saw his name come up in my newsfeed until I read the end of the tweet.
"I think it’s just really rude and disrespectful, there’s no need to respond with that kind of attitude, and the fact that he can’t even apologise. And why block me? I didn’t do anything wrong."
She added: "I'm still absolutely fuming. It was so cruel.
"He’s not just offended me, but a lot of plus sized people all over Britain."
Sir Alan Sugar has so far declined to comment on the incident.