Willows Dental and Implant Centre in Wombourne posted a video on Facebook showing a drone flying through the sky carrying a box.
The post said the antibiotics were delivered on May 1 within seven minutes with the drone flying at 27mph.
But the practice then shared an apology on its page after receiving a visit from the police who shared "words of advice".
Drones are legal in the UK but operators must be registered and the aircraft must have permission from the Civil Aviation Authority to fly near towns and cities.
The post said: "As a dental practice that's constantly pushing the boundaries in patient care, it seems in carrying out drone deliveries of essential medicines to clients in need in a time of need and lockdown, we upset some people on social media that take drone flying very seriously and reported us to the police.
"We can assure you that no one was hurt, our patient is out of pain and it wasn't our intention to upset anyone.
"We genuinely thought we were helping people in lockdown, and didn't think we'd be interfering with any planes in the sky.
"So for those of you that reported us to the police, consider our knuckles rapped. We're sorry, for thinking too far outside the box."
Dr Neel Barchha, from the Willows practice, told the Express & Star: "The incredibly professional police officers from Wombourne are satisfied that no crimes have been committed, a risk assessment was carried out and no rules and regulations were broken.
"May we take this opportunity to apologise to the professional drone community for any offence caused and wish the people of Wolverhampton the best of health over this difficult time.
"We continue to serve our community and carry out essential medical and dental care without the use of drones."
A Staffordshire Police spokesman said: "Officers were notified of a social media video of a drone and a post from a man stating he was delivering drugs to patients from a dental surgery in Wombourne at noon May 9.
"Officers attended the man’s address and words of advice were given."