John Dramani Mahama, former president of Ghana, urges all sides to exercise restraint a day after election-related violence killed three people in Harare, the capital.
Mnangagwa, whose ruling ZANU-PF party has already secured a parliamentary majority, called for unity and peaceful settlement of differences.
Crisis in Zimbabwe Coalition, the country's largest group of non-governmental organisations, said the July 30 election "falls short of a credible process".
Zimbabwe's former finance minister claimed that it was a "fact" that he and MDC leader Chamisa, 40, are on a "hit list", without giving evidence of his assertion.
MDC Alliance supporters demonstrate outside the Zanu-PF headquarters.
The military remains on high alert in Harare as the final results from Zimbabwe's presidential election - the first in 37 years - are being tallied.
On the streets of the capital Harare, we found supporters and party members who were so frustrated, they could barely get the words out.
The protestors had come out in large numbers to oppose the House of Assembly election results that gave the ruling Zanu-PF party a more than two-thirds majority, and were now demanding the release of the presidential results, which they accused ZEC of tampering with, to deprive opposition leader Nelson Chamisa of the MDC Alliance, a clear victory.
The government says the army was deployed in central Harare to help police restore order.
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"We hold the opposition MDC Alliance and its whole leadership responsible for this disturbance of national peace, which was meant to disrupt the electoral process".
Earlier the ZEC said it was still not ready to release the presidential results, but it would know later on Thursday exactly when those results could be made public.
This photo taken on January 7, 2017 shows Zimbabwe's then acting President Emmerson Mnangagwa speaking during a funeral ceremony in Harare.
The MDC also questioned the pace of releasing results from the presidential vote.
He said: "We have seen how we have won in those areas including rural areas, we were leading, and now they want to hide on the Zanu PF usual manipulation that they control rural areas". At one point, the relatives blocked hospital staff from wheeling the body to the mortuary and demanded a police explanation; a plainclothes officer said they could return Thursday to pick up the body after a police investigation.
Mnangagwa, 75, had promised a free and fair vote after the military ushered him to power in November when Mugabe was forced to resign.
Questioning the independence of the judiciary, Chamisa said he was reluctant to go to court to challenge the results, saying this would be "going into the lion's den".
If no presidential candidate wins at least 50 percent of the ballots cast in the first round, a run-off vote is scheduled for September 8.
The outcome will be announced once results from all provinces are received and verified, Priscilla Chigumba, the chairwoman of the electoral commission, told reporters.