The mother of five was taken from her home in the middle of the night on 27 December alongside her husband and a niece.
Her husband was released in March after a ransom was reportedly paid but the kidnappers held onto the ophthalmologist and her relative.
The doctors say they will not even provide emergency care during the strike.
Dr Popoola works for the National Eye Centre hospital in Kaduna, north-west Nigeria, and lives in the official quarters provided by the hospital.
The hospital is one of the biggest eye hospitals in the country.
Experts say the hospital's location on the outskirts of Kaduna city makes it an easy target for kidnappers.
In 2021, dozens of students were taken from the nearby college of forestry.
Dr Taiwo Shittu of the Lagos University Teaching Hospital said what happened to Dr Popoola could happen to anyone.
“We want the authorities to act fast, this has dragged on for too long,” he said in a video on social media.
The doctors feel the security agencies are not doing enough to secure her release.
The kidnappers are asking for 40m naira (£19,000; $25,000) for their freedom.
Although a controversial law criminalising ransom payments came into effect in 2022, they are still often paid by relatives desperate to free their loved-ones.
The law carries a jail sentence of at least 15 years for anyone who pays a ransom, although no-one has yet been convicted.
The government is yet to comment on the strike or the doctor’s situation.
President of the doctors' association Dr Dele Abdullahi told BBC News that “the family was exploring a diplomatic route initially, but they have now given us the permission to explore other options”.
The doctors' association recently held a march in public hospitals across the country and gave the government a two-week ultimatum for “the unconditional release” of Dr Popoola.
In recent years, kidnapping has become rife in Nigeria, with hundreds of people abducted, largely by criminal gangs who see it as an easy way to make money. It has been particularly bad in the north-west of the country.
By Azeezat Olaoluwa & Mansur Abubakar, BBC
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