Last week, Draghi lashed out at Erdogan for hosting a summit with the EU's two top leaders at which European Commission chief Ursula von der Leyen ended up without a chair.
Turkish officials rejected the blame for the incident, saying they were applying protocol instructions received from Brussels.
But it turned into the main talking point of the first Turkish-EU summit in a year, and was branded "sofagate" on Twitter, overshadowing attempts by both sides to mend months of rocky ties.
Speaking to reporters in Rome, Draghi said he was "sorry for the humiliation that the president of the commission had to suffer with these, let's call them for what they are, dictators".
In his first public comments on the episode, Erdogan said Draghi's call jeopardized Turkish-Italian relations, AFP reported.
"The comment made by the Italian prime minister is tantamount to complete impertinence, a complete vulgarity," Erdogan said in a video clip posted on Twitter.
"At a time when we hoped relations between Turkey and Italy would reach a good point, this man called Draghi unfortunately scuppered the ties."
Erdogan also urged Draghi -- who he pointed out had been appointed and not elected by the people -- to look back at Italy's history, in an apparent reference to the reign of fascist dictator Benito Mussolini.
"You should first of all be aware of your history to make such a comment about Tayyip Erdogan, but we see that you are not," the Turkish leader said.
Draghi's comments sparked a furious immediate response from Ankara, which called in Italy's ambassador for an explanation within an hour of the Italian premier's remarks.
But the episode also created a scandal in Brussels, where European Council President Charles Michel came under scathing criticism for taking one of the two seats and relegating von der Leyen to a nearby sofa.
Michel has spent much of the past week apologising, saying he has spent sleepless nights thinking about how he could have handled things better.
The two EU leaders explained their versions of events to the European Parliament's political groupings on Tuesday, with Michel reportedly stressing the importance of moving on.
"He insisted on EU unity and team spirit," and "the need to move on and focus on the EU's priorities and challenges," a participant in the meeting said.
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