Vemić turned professional in 1995. He reached a career-high singles ranking of world No. 146 on 25 February 2008 and a career-high doubles ranking of No. 31 on 12 January 2009. He is ranked 19th on the list ofFastest recorded tennis serves with a serve speed of 235 km/h (146 mph).
In 2007, Vemić began the season ranked No. 406, a drop of 170 places from early 2006. That year, he competed inFutures, theChallenger Tour, and selectedATP Tour events. His results included two Futures semifinals, one final, three ATP main draw qualifications, and a win in a top-level Challenger. These results returned him to the top 300 in April. He later reached the semifinals of the Challenger inBermuda and qualified for the main draw of theFrench Open, where he lost in four sets toJanko Tipsarević. In August, he reached the semifinals inSegovia and the final inBinghamton, returning to the top 200 for the first time since 2004.
He won five doubles titles on the 2008 ATP Challenger Tour: inMiami, United States (with Ilija Bozoljac);Sunrise, Florida (with Janko Tipsarević);Cremona, Italy (with Eduardo Schwank);Waco, Texas (with Alex Bogomolov Jr.); andCalabasas, California (with Ilija Bozoljac).
In Grand Slam competition, Vemić reached the doubles semifinals of the2008 French Open and the quarterfinals of the2008 US Open, partnering with Bruno Soares.
Following his professional playing career, Vemić briefly coached WTA playerAndrea Petkovic, assisting her and coach Petar Popović in reaching a career-high WTA ranking of No. 9.
Soon afterward, Vemić joined the coaching team of world No. 1Novak Djokovic. As part of Djokovic's team, he coached alongside longtime head coachMarián Vajda during several major tournaments, including theUS Open (2011, 2012), theAustralian Open (2012, 2013), theFrench Open (2012, 2013), andWimbledon (2012, 2013). During this period, Djokovic won two Grand Slam titles at the Australian Open and one at the US Open, as well as the 2012 ATP World Tour Finals in London.
In 2016, Vemić coached Djokovic at the ATPMiami Open, where Djokovic defeatedKei Nishikori 6–3, 6–3 in the final, surpassingRoger Federer in career prize money earnings.
In January 2013, Vemić joined the coaching staff of theSerbian Davis Cup team, helping lead the team to theDavis Cup final against theCzech Republic. He continued as part of the coaching staff during the2016 Davis Cup World Group competition.
In August 2016, Vemić served as the official coach of the Serbian men's tennis team at the 2016 Summer Olympics in Rio de Janeiro. The team included world No. 1 Novak Djokovic, former world No. 1 doubles player Nenad Zimonjić, and top-30 ATP player Viktor Troicki.
From August 2016 to 2017, Vemić was the head coach of the American doubles teamBob and Mike Bryan, considered the most successful doubles team in tennis history.