MOSCOW, Jan. 27— The Soviet authorities have abandoned construction of a nuclear power plant near the Black Sea in the face of public opposition generated by the 1986 accident at the Chernobyl atomic power station, a Soviet newspaper reported today.

The newspaper, Komsomolskaya Pravda, also disclosed that all of the 20 operating nuclear power plants in the country and most of those under construction were now ''bitterly opposed'' by local residents.

The report on the abandoned project near the city of Krasnodar was the strongest official acknowledgment that since the Chernobyl accident public opinion here had turned sharply against nuclear power and posed a serious threat to the country's ambitious nuclear energy program. Protests Reported Unofficially

''The Krasnodar nuclear power plant is perhaps the first to witness this, but it is not the only one,'' the newspaper said.

Public protests against the siting of nuclear power plants have been described in underground publications, but the controversy has only been hinted at in the official press.

In late November the head of a Soviet commission investigating the Chernobyl disaster told a Frankfurt environmental magazine that public protests had blocked plans for new nuclear plants near Minsk, the capital of the Byelorussian Republic, and near the Black Sea port of Odessa.

If that is so, it has not been officially announced here. The commission official spoke during a visit to West Germany. 'Unprecedented' Action

Komsomolskaya Pravda said the abandonment of the Krasnodar project was ''unprecedented'' and had caused nuclear power officials to fear ''a chain reaction'' that would threaten other projects.

Since the Chernobyl accident, Soviet officials have repeatedly declared their intention of pushing ahead with a construction program that is meant to double nuclear power output by the year 2000. At that point, according to the official timetable, nuclear power would account for more than 20 percent of the country's electricity.

The Komsomolskaya Pravda report said $41 million had already been spent on the new Krasnodar plant before local officials bowed to a flood of letters from local residents and decided at the end of last year to cancel it.

The newspaper did not provide the precise location of the plant, the state of construction or details of its planned output and type. The plant is not mentioned in recent Soviet reports, indicating it must be quite new. Like officials in many Soviet localities, the newspaper said, leaders in the Krasnodar region had pleaded with Moscow two years ago for financing to build a nuclear plant to meet an expected shortage of electricity in the northern Caucusus Mountains.

''Before Chernobyl, to have a nuclear power plant was profitable - and also prestigious,'' the newspaper said. ''In addition to providing self-dependency in energy, a nuclear power plant guaranteed increased attention toward the region, including jobs, housing and social and cultural amenities.

''But then the entire world started talking about Chernobyl.''

Thirty-one people died when the No. 4 reactor at Chernobyl exploded and burned as the reactor raged out of control during an experimental shutdown of the cooling system. The explosion rained radioactive particles on surrounding Ukrainian towns and farmland, and contaminated farmlands as far away as Poland and Sweden.

The accident forced the evacuation of more than 100,000 residents.

Until today's report, the explosion seemed, in the view of Western analysts, to have set back Western nuclear power programs more severely than the Soviet program. 'Situation Is Not Surprising'

The shock of Chernobyl brought new determination to anti-nuclear groups in the United States and Europe.

''I think the situation today is not surprising,'' the Komsomolskaya Pravda reporter wrote. ''We have read about the anti-nuclear demonstrations in other countries, and about the referendum in Austria in which it was decided not to build nuclear power plants at all, and about the fears connected with this in the United States.''

The Soviet authorities have worked intensively since the Chernobyl accident to assure the public that nuclear power is safe and that their program would stay on course.

The three remaining Chernobyl reactors have been brought back on line, and new safety measures have been adopted at other nuclear power plants, including installation of more safety equipment and intensive training courses for plant employees.

Soviet officials have announced that they would not build any more reactors of the type that failed at Chernobyl. The reactor model uses graphite to ''moderate'' or help sustain the fission of its nuclear fuel. Most nuclear power plants in the United States are of a different design in which water moderates the nuclear reaction. Swedes visiting Moscow this week were reportedly told that plans for seven graphite-moderated reactors had been scrapped, including the fifth and sixth reactors for Chernobyl.

Komsomolskaya Pravda, the newspaper of the Young Communists' League, argued that critics of nuclear power were misguided. The paper said that the northern Caucasus would face a shortage of eight million kilowatts of electricity by the year 2000, and that building coal-burning power plants instead, as residents propose, would bring pollution to the fertile Kuban Valley.

Map of Soviet Union showing location of Krasnodar (NYT)