Bhupathi-Bopanna in semis, Sania-Bethanie bow out in Rome
Berdych, who shocked world number one Novak Djokovic in the men's singles in a gruelling three-setter, cited muscle fatigue as the reason for him withdrawing.
Bhupathi and Bopanna will now fight it out with Mexico's Santiago Gonzalez and American Scott Lipsky for a place in the final.
Gonzalez and Lipsky came from a set down to upset second seeded Spaniards Marcel Granollers and Marc Lopez 1-6, 6-4, 10-6 in their quarter-finals.
Meanwhile, Sania and Bethanie's campaign ended in the women's doubles after their 4-6 3-6 defeat against top seeds and local favourites Sara Errani and Roberta Vinci.
The Indo-American pair lost in one hour and 15 minutes as they squandered six of the nine break chances.