A renowned shot blocker, Vranković made little impact in the NBA, managing small minutes and a low scoring average even though he started 73 out of 170 games played. He did however have a major impact in Europe playing for the Greek League team Panathinaikos. In 1996, he combined with Dominique Wilkins to lead Panathinaikos to a Euroleague championship - a first for the club. Vranković's defensive contribution was vital, namely his block of Zeljko Rebraca at the buzzer, in the third game of the quarter-finals against Benetton Treviso, sending Panathinaikos through. In the Euroleague Final, his block to deny Barcelona the lead in the last second, gave the game a dramatic end, before falling to the ground in the ensuing hustle and clutching his knee in pain at the buzzer. In an interview at the official Euroleague Final Four website, Panathinaikos captainPanagiotis Giannakis described the incident as follows:
"All that happened at the end of the game is unbelievable, a few seconds that lasted one century! I remember Stojko (Vrankovic) running like...Carl Lewis from one side to the other to stop (José Antonio) Montero. He blocked the layup almost at the buzzer and he sealed the victory."
Vranković was born in Drniš to a Serb father and Croat mother, and was brought up by his mother. He identifies as Croat, and was present with Dino Rađa at the Knin fortress when President Franjo Tuđman held the speech after Knin had been taken by the Croatian Army (1995). According to his neighbours in Drniš, at this time, he had his father's remains reburied from the Orthodox cemetery to the Catholic cemetery.
Vranković was a close friend of fellow player Dražen Petrović. He was a coffin bearer at Petrović's funeral in 1993, and has been active in commemorating his memory. In an article at the FIBA website concerning the opening of a museum in Dražen's honour, Vranković was quoted as saying: "I would like to thank all those involved in creating this place, so children would be able to learn more about 'basketball's Amadeus'".