The Bulgarian also appeared at the Serbian leg of the tour last week and has come into contact with top 10 players Novak Djokovic, Alexander Zverev and Dominic Thiem.
He returned home to Monaco after feeling unwell during his loss to Coric, and has since returned a positive test. The Zadar final between Novak Djokovic and Andrey Rublev has now been cancelled.
Unvaccinated players could be allowed to play at Australian Open
However, there is no indication of when Dimitrov picked up Covid-19.
"I want to reach out and let my fans and friends know that I tested positive back in Monaco for Covid-19," Dimitrov wrote on Instagram.
"I want to make sure anyone who has been in contact with me during these past days gets tested and takes the necessary precautions.
"I am so sorry for any harm I might have caused. I am back home now and recovering. Thanks for your support and please stay safe and healthy."
Dimitrov was filmed dancing in a nightclub with Djokovic, Zverev and others after last week's Adria Tour event in Belgrade, while he also joined his Zadar co-stars for a game of basketball on Thursday.
Highlights: Grigor Dimitrov struggles in Borna Coric defeat
With international tennis suspended due to the COVID-19 pandemic, men's world number one Djokovic organised the Adria Tour as a charity event designated to be contested over four legs across the former Yugoslavia.
Austria's Thiem won the opening leg in Belgrade, where a capacity 4,000 crowd turned up on each day at Djokovic's tennis centre by the Danube river.
With both Serbia and Croatia easing lockdown measures weeks before the event, players were not obliged to observe social distancing rules in either country and were seen embracing at the net at the end of their matches.
Fans in Croatia, however, were subject to the rules and had to keep their distance, meaning that the 9,000-capacity Visnjik tennis complex was half-full for the event.
Novak Djokovic defends 'phenomenal' Adria Tour
The third leg due in Montenegro on June 27 and 28 was scrapped last week over coronavirus concerns while the final leg, scheduled for July 3 and 4 in Bosnia's Banja Luka, is now likely to be called off.
Djokovic's brother Djordje, the director of the entire Adria Tour series, said everyone who has been in contact with Dimitrov would be tested for COVID-19 in the next few hours.
"After receiving the news, we took all the measures stipulated by the Croatian government and everyone who has had any contact with Grigor will be tested," he said in a courtside interview.
With additional reporting from Reuters
'I liked what Novak said' - Medvedev backs Djokovic over vaccine being 'personal'
Opinion: Djokovic facing big decision, but don't make him the villian again