The first of up to four planned flights to Wuhan had been due to depart South Korea on Thursday morning, but were delayed for unspecified reasons, Yonhap news agency reported. The flights were still expected to depart by later in the day.
"The weapons that will protect us from the new coronavirus are not fear and aversion but trust and cooperation," Moon said in a speech on Thursday, decrying "fake news" as creating excessive anxiety.
Protesters used tractors to block access to facilities earmarked as quarantine centers in the cities of Asan and Jincheon, about 80 km (50 miles) south of Seoul, on Wednesday.
The government has proposed to keep the evacuated citizens in isolation at the facilities, usually used as training centers for public servants, for at least two weeks to ensure they do not develop symptoms.
Moon tried to reassure residents, saying the initial evacuees would only be those with no symptoms, and they will be held in isolation.
"The government will be taking air-tight steps to ensure the residents of areas where the facilities are do not need to be concerned," he said.
South Korea has reported four cases of the newly identified virus, all of whom arrived after visiting the central Chinese city of Wuhan where the pathogen appeared in humans late last year.
The new strain of coronavirus has an incubation of between one and 14 days and can spread before symptoms show, complicating efforts to screen people at international borders.
It has created widespread alarm because it is spreading quickly and there are still important unknowns surrounding it, but it is still too early to know what its death rate will be, because there are likely to be many cases of milder infections going undetected.
South Korea said the inter-Korean liaison office in Kaesong, North Korea, would be closed until the outbreak eased. North Korea would not allow South Korean officials to enter the office when they arrived on Thursday, Yonhap reported, citing South Korea's unification ministry.