The "unprecedented" exercise will be launched in the next few days, with at least 10,000 field workers deployed across the country of 57 million people, he said in an address to the nation.
"We are now entering a new phase in the fight against the coronavirus pandemic," said the president.
"In the coming days, government will be rolling out a screening, testing and tracing and medical management programme on a huge scale."
Field workers will visit homes in villages, towns and cities to screen residents, and people with severe symptoms will be hospitalised.
Those infected but asymptomatic or with moderate symptoms will have to remain in isolation at home or at government quarantine centres, the president said.
Mobile technology will be used to trace people who will have been in contact with confirmed coronavirus patients.
"This drive is far-reaching, it is intensive and it is unprecedented in scale," he added.
The numbers of confirmed infections had risen to 1,326 cases with three deaths, he said.
South Africa was Monday observing its fourth day of a 21-day lockdown order, but authorities have struggled to keep everyone at home.
Ramaphosa said most people had "responded responsibly" to the lockdown.
But he expressed concern about those who had defied the order and "have not yet appreciated the seriousness of this disease".
Police say they have made hundreds of arrests so far of people violating the lockdown