Iranian state TV warned there could be more explosions.
Authorities initially said that 13 people were killed and six others injured in the blast at the Sina At'har health center on June 30.
But Jalal Maleki, spokesman for the Tehran Fire Department, was quoted as saying that firefighters recovered six more bodies after the ensuring blaze was extinguished.
The dead included 15 women and four men, Maleki said, adding that firefighters had rescued 20 people.
The spokesman said the explosion occurred as gas canisters caught fire in the clinic's basement.
Deputy Tehran Governor Hamidreza Goudarzi earlier said that a leak from medical gas tanks in the building was the cause of the explosion.
Tehran prosecutor Ali Alqasimehr said that those injured were transferred to Tajrish hospital for medical treatment.
Ambulances were deployed to the site while police blocked off all roads leading to it.
Video posted online appeared to show more than one explosion and thick black smoke rising into the sky.
State TV reported there could be more blasts because there were a number of oxygen tanks remaining in the clinic.
The incident comes four days after the Defense Ministry said an explosion occurred at a gas-storage facility to the east of Tehran due to "leaking gas tanks."
No casualties were reported after the blast, which occurred in an area with a sensitive military site.
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