Federer, 37, a four-time finalist in Rome, was competing at the event for the first time since 2016.
Tsitsipas will face Fernando Verdasco or Rafael Nadal in the semi-finals.
“I am disappointed that I will not be able to compete today. I am not 100% physically and after consultation with my team, it was determined that I not play,” said 20-time Grand Slam champion Federer.
“Rome has always been one of my favourite cities to visit and I hope to be back next year.”
Spain’s defending champion Nadal, chasing a ninth Italian Open title, is in quarter-final action against compatriot Verdasco later on Friday.