With both Bhambri and Ramanathan bowing out, Sania Mirza, who is in her last year on tour, remains the only Indian confirmed to play at Wimbledon this year.
Sania and Lucie beat Italian duo of Jasmine Paolini and Martina Trevisan, 4-6 6-2 6-1 in their opening round.
He combined with American Hunter Reese to win 7-6(4) 6-3 in their opening round.
Bopanna and Ramkumar defeated top-seeded Ivan Dodig of Croatia and Brazil's Marcelo Melo 7-6 (6), 6-1 in the final.
Both Prajnesh Gunneswaran and Ramkumar had lost their singles matches on Friday and to stay alive in the tie, the Indians needed to win the doubles.
Prajnesh Gunneswaran yet again lost to a lower-ranked player in the Davis Cup while Ramkumar Ramanathan went down fighting in the second singles.
It was the first time in open era that two Indian pairs competed against each other at a Grand Slam.
Ramkumar Ramanathan will play for Mumbai Leon Army. He was picked up for Rs 4.5 lakh.
The unseeded Indian, celebrating his 26th birthday, lost the summit clash 4-6, 4-6 to his seventh seed Sebastian Korda, in one hour and 23 minutes.
In a mismatch of epic proportion, Ramkumar double-bageled 17-year-old Muhammed Shoaib in just 42 minutes in the opening singles.