Gabashvili began playing tennis at the age of six in Tbilisi, Georgia. His mom Anna, who is a doctor, took him to play tennis and began coaching him until the age of ten. Gabashvili's family moved to Moscow, Russia because there he had better opportunities to pursue his dream of becoming a pro tennis player. Andrei Kesarev began coaching him when he was ten years old and at the age of 18 he turned pro. He's nicknamed "Tsunami" because of his energy.
In the first round of the 2007 US Open, Gabashvili defeated World No. 7 Fernando González in five sets. In the fourth set, Gabashvili served for the match at 5–4, but hit three consecutive double faults. He lost the game and the set, but came back in the fifth and final set to win the match.
Gabashvili represented his country at the 2011 Summer Universiade held in Shenzhen and won a silver medal. Despite being ranked below the top 100 in the ATP rankings, Gabashvili was still by far the highest-ranked player in the singles draw, and was thus a favorite to win Gold. He advanced all the way to the singles final without dropping a set before suffering a surprising defeat at the hands of Lim Yong-Kyu, an established member of South Korea's Davis Cup team.
At the 2015 Citi Open, Gabashvili upset two time Grand Slam champion and world No. 3 Andy Murray in the second round in three tight sets to claim only his fourth ever win against a Top 10 player, however he lost to Ricardas Berankis in the next round in two sets.
Gabashvili speaks Russian, Georgian, Spanish and English. He has a daughter named Nicole. His younger sister Ekaterina graduated from Vanguard University of Southern California. His younger brother, Levan played professional basketball for Dinamo Tbilisi, until he started to attend college at Seward County Community College. Outside of tennis, Gabashvili enjoys spending time with his daughter, and also likes reading, playing football, and basketball. He is coached by Guillermo Cañas.