Wozniacki and Giorgi were interrupted in the second set when a spectator was removed from the stands on a stretcher as Giorgi led 4-3 on serve. When play resumed, Giorgi failed to win a break point to lead 5-3 and Wozniacki went on to win three straight games to level at one-set all.
Wozniacki remained under pressure in the final set, fighting off two break points before breaking to lead 2-1. A double-fault at 4-2 left Wozniacki serving for victory, and in sight of her first title since October.
"It feels good," Wozniacki said of her win. "Game feels in really good shape. Camila played really well. I really had to be focused out there."
Kerber fought off four break points just to hold for 1-1 before taking charge against Makarova, breaking twice to take the set. In the second set, she needed four break points to win the first game before once more dominating Makarova.
"It was not as easy as the score said, because there were tough rallies," Kerber said. "She’s a great player on grass, so I was just focusing on me, trying to be aggressive and take my chances. I did a good job."
Kvitova’s withdrawal gave British wild card Heather Watson a walkover into the semifinals.
Watson will meet Madison Keys, who beat fellow American Lauren Davis, a qualifier, 6-2, 6-1. Wozniacki faces Kerber.
Men’s top seed Richard Gasquet also reached the semis, beating Slovak Martin Klizan 6-3, 6-4. It is his third semifinal of the year and comes after he was sidelined for more than three months with a back injury.
The Frenchman, who claimed the title in 2005 and 2006 when the event was held in Nottingham, broke for 3-1. That was enough to earn him the opening set, and in the second set he again fought off a break point before breaking to lead 4-3.
Gasquet is playing only his third tournament since March, and admits that although he is progressing he is still not fully fit.
"It’s always difficult to come back," Gasquet said. "You are questioning yourself every day on your body. I’m not 100 percent on my game and everything, but I’m playing better and better. I served well. I could play with power."
Gasquet will next play Denis Istomin of Uzbekistan, who beat Edouard Roger-Vasselin of France 7-6 (5), 6-2.
Spanish defending champion Feliciano Lopez will play his quarterfinal against Jeremy Chardy of France on Friday after the match was canceled overnight due to rain. The winner will play Sam Querrey, also on Friday, after the American beat Frenchman Julien Benneteau 7-6 (5), 6-4.