Peter Magombeyi, president of the Zimbabwe Hospital Doctors Association (ZHDA), was one of the organizers of an ongoing strike to demand higher wages for state doctors because of soaring living costs, before he disappeared on Saturday night.
ZHDA represents mainly junior doctors at public hospitals.
His colleagues have alleged he was abducted by the security services in an attempt to break the strike, which has disrupted healthcare services in hospitals, but a senior information ministry official denied the government was responsible.
Rights groups accused the government of abducting activists in the lead-up to planned opposition protests last month, when the government also denied responsibility.
Mnangagwa's government has said it will not use excessive force against protesters after being criticized over its heavy-handed response to post-election violence last year and protests over fuel price hikes this year.
Doctors chanted "No Peter, no work" and held placards reading "Bring Peter back" as riot police stopped them from marching beyond the High Court building.
The doctors dispersed after police allowed half a dozen of them to deliver a petition to Mnangagwa's office demanding action over Magombeyi's disappearance, but they said their strike would continue.
"Dr Peter Magombeyi's whereabouts remain unknown, and we are seriously concerned about this. We therefore demand his unconditional return to his family," the petition read.
Magombeyi's sister on Sunday filed an urgent application in the courts to compel the government to do everything to find her brother. The court was due to hear the case on Monday.
Police could not comment on the matter.
Two years after Mnangagwa and the army conspired to oust longtime ruler Robert Mugabe, people are suffering from triple-digit price rises and shortages of basic goods like fuel.
Mnangagwa's government has promised salary increases to public sector workers, but many say those rises are insufficient.
Diplomats said they were monitoring whether authorities respect the people's right to protest peacefully.Some are skeptical since Mnangagwa was Mugabe's trusted lieutenant at times when the former president, who died this month, crushed dissent during his near-four decades in power.