Sir Stelios Haji-Ioannou said in a statement that he had issued the libel writ in a bid to protect his reputation and that he would seek "substantial damages" for the allegation that he was a liar, according to news agency Reuters.
"With this advert he went too far and I am obliged to draw the line here. This is a libel action about my own reputation as an individual, not about comparing the two airlines," he said. "He called me a liar. I am not a liar. That is libellous, and I will seek substantial damages."
EasyJet has also threatened Ryanair with legal action over comments made about its pricing strategy in a BBC programme. That spat centres on whether it was legitimate to call easyJet a "'high fares airline" and whether the airline forced customers to pay for extra items.
Consumers have long complained that budget airlines such as Ryanair and easyJet have added too many extras on to originally-advertised fares. Consumer regulator the Office of Fair Trading is investigating 'drip pricing' strategies on retail websites, including those for airlines.
The Government will shortly begin a consultation on what penalties to apply to airlines which break EU rules on airlines' hidden charges.
Pugnacious O'Leary is famed for his publicity-grabbing stunts and comments and his adverts attacking those he sees as the company's enemies. In the past these have included airport authorities, regulators and government ministers as well as competitors.
O'Leary first offered to settle the dispute by a run around Trafalgar Square in London. After news of the libel suit became public he changed his offer.
"Stelios obviously can't run, but he can't hide either," O'Leary was quoted in the Ryanair statement as saying. "If he's too scared to run then why doesn't he accept our challenge to a bout of Sumo wrestling instead."
Ryanair is now allowing visitors to its website to vote on what dualling activity should settle the dispute, with other options including a drag race and a wheelbarrow race.
Haji-Ioannou is easyJet's founder and largest shareholder, but does not have an executive role at the company. He sits on its board but is not the firm's chairman.
Ryanair's original advert challenged easyJet to publish its punctuality statistics and depicted Haji-Ioannou with a long, Pinocchio-style nose.
Haji-Ioannou's lawyers wrote to Ryanair saying that the implication that he was a liar and was suppressing the statistics when he in fact had no executive role at the company was defamatory.
In response O'Leary wrote to Haji-Ioannou's lawyers saying that their claims were "full of rubbish" and promised to continue to run the ads until easyJet published the punctuality statistics.
