Elia was angered by Lucio's part in an incident on Sunday where the Brazilian's trailing leg caught Dutch compatriot Maarten Stekelenburg on the face which left the Roma goalkeeper unconscious.
The 24-year-old winger posted dn his Twitter account: "i hate lucio #inter he kick #stekelenburg in the face he have to get a red card :@ i am really angry."
In response to Elia's post, Lucio told Mediaset that the former Hamburg player's comments were "stupid".
"It's only normal, because now he's a Juventus player. I would never say something like that. I don't hate anybody," he said.
"I don't think it's right to even talk about it, because not even he can say or know what happened on the field.
"I don't know why he said it, but this isn't important for me. The only important thing is that Stekelenburg is all right.
"As I already said, it was an involuntary situation. What Elia said is stupid."
The offending tweet has already been deleted.
Stekelenburg was kept in hospital overnight but was released after a scan gave him the all-clear, although he did need two stitches for his head wound.
He is not expected to return to action for Roma when they face Siena in midweek Serie A action.