The Bulgarian, 29, withdrew from the second tournament in Zadar, Croatia on Saturday with sickness after his opening loss to Borna Coric.
"I am so sorry for any harm I might have caused," he said.
Sunday's final, which would have featured world number one Djokovic, has been cancelled.
Dimitrov, who also played in the first event in Belgrade last week, wrote on Instagram: "I tested positive back in Monaco for Covid-19. I want to make sure anyone who has been in contact with me during these past days gets tested and takes the necessary precautions.