An American from a cruise ship visiting this lovely Caribbean island for a few hours the other day had two items on his agenda. First he wanted to eat some turtle stew, a local delicacy. Then he wanted to taxi out with his wife to photograph the new airport, being built by Cubans and denounced by President Reagan last month as a threat to U.S. security.

The tourist could do both with equal ease--only two weeks after Prime Minister Maurice Bishop warned that a U.S.-backed invasion was "imminent" and Grenadans were urged to dig trenches in the white sand beaches or guard against a parachute attack from U.S. planes in the verdant inland hills.

The American's relaxed excursion was one man's measure of the gap between what is being said officially in Washington and St. George's about the four-year-old Grenadan leftist revolutionary government and what appears to be taking place on the 133-square-mile island--whose 110,000 residents make it a little more populous than Alexandria, Va.

The People's Revolutionary Government and the Reagan administration both point to precedents in Cuba and Nicaragua, the region's other chief revolutionary powers, as the basis of their concern.

U.S. diplomats cite Soviet military facilities in Cuba and the recent talk of a possibility of Soviet missiles being put in Nicaragua, along with the support of Soviet-aligned revolutionary movements in other nearby countries. Bishop and his followers cite U.S. backing for the Bay of Pigs invasion against Cuba 22 years ago and the present U.S. backing for Nicaraguan counterrevolutionaries fighting to overthrow the Sandinista government in Managua.

So, both sides say, similar things are about to happen in this little country in the chain of Windward Islands, sitting in the Caribbean sun about 150 miles north of Venezuela and astride critical routes for shipping oil to the southern rim of the United States.

Consulted Grenadans, good-natured and friendly, did not seemed alarmed by Bishop's warnings of an invasion. One noted that officials began playing them down after Bishop and most of his Cabinet attended an important cricket match against a visiting Indian team and free-spending tourists from Trinidad started arriving for an annual sailing regatta. Then Bishop left for a week-long visit to North Korea.

Reagan, displaying an aerial spy photo on nationwide television, pointed at the airport under construction at Point Salines just southwest of St. George's as of particular concern to the United States. The runway--Reagan said it was 10,000 feet while Grenadans say 9,000--was described by then-secretary of state Alexander M. Haig Jr. as a potential base for "every aircraft in the Soviet-Cuban inventory" and by Reagan as a menace to U.S. security interests.

Cuba is supplying 40 percent of the estimated $71 million the airport will cost, much of it in equipment and labor. About 400 Cubans are at work laying tarmac and building a terminal. A group of them was seen playing baseball last week, just below new Cuban-built living quarters that U.S. officials fear will turn into permament Cuban barracks after completion of the airport, scheduled for mid-1984.

Nestor Sanchez, U.S. deputy assistant secretary of defense for Inter-American affairs, said in February that U.S. security officials also are concerned about a military camp and "naval facilities" at Calivigny Point near the airport. He said the air and sea installations "far exceed the requirements of that tiny island."

Here on Grenada, however, the airport is regarded as the key to invigorating a sagging tourist industry since it will be able to handle large jets full of tourists. The island's present airport at Pearls is an hour away from the main beach hotels near St. George's. Its runway can accommodate only small planes and it cannot handle night flights, creating what Grenadans call the "overnight trap" that requires tourists to spend the night and change planes on other islands.

Royston Hopkin, head of the Grenadan Hotel Owners' Association, said the 18-passenger planes serving Pearls cannot carry in enough passengers to provide an adequate occupancy rate for the island's 200 hotel rooms and 300 guest-house rooms. He recalled that Barbados and other neighboring islands have airports as big or bigger than the one underway here. Reagan's description of the project as a potential military threat, he added, is "baloney, a lot of baloney."

Reagan's display of intelligence photos taken by satellites or overflights provoked derision here. Grenadans often take a spin out to the unguarded site to see how the Cubans are progressing and tourists can take photos with their Instamatics. U.S. students at the St. George's Medical School here race motorbikes on the unfinished runway and can sputter among the Cuban installations without hindrance.

The Calivigny camp, easily viewed from neighboring hillsides, contains garages, sheds and a rifle and training range where nearby residents say Grenadan soldiers can be heard firing occasionally at an island just offshore. As far as can be seen, the naval facility is limited to an undeveloped inlet that, according to British Admiralty charts and a Florida firm that dredged last year, measures about 20 feet deep.

Deputy Prime Minister Bernard Coard said in an interview that the government eventually plans to turn the little bay, Egmont Inlet, into a yachting harbor because it offers natural protection from the weather.

Bishop's government has made public and private pledges that it intends the airport only for tourism and trade. In a letter to Reagan dated Aug. 11, 1981, Bishop said he and his followers "have no intention, and have never had any intention, of using this modest international airport for any purpose other than as the means of developing our relatively untapped tourist potential and for the development and expansion of regional and international trade in fresh fruits and vegetables and agro-industrial products."

James Emmanuel, permanent secretary of the Grenadan Foreign Ministry, said the letter was delivered to the U.S. Embassy in Barbados, which serves the surrounding islands, but never received an answer. A letter dated March 26, 1981, also went unanswered, he said, part of a diplomatic boycott in which the United States refuses to accredit a Grenadan ambassador and bars its own ambassador for the area, Milan Bish, from visiting Grenada.

Reagan administration officials have since insisted that, whatever his intention now, Bishop's alignment with Cuba and the Soviet Union means the airport will give Cuban and Soviet aircraft a strategic convenience they previously did not enjoy. Similarly, they say, the Cuban-Soviet tie opens the possibility of Soviet monitoring equipment being installed here to keep track of U.S. submarine movements in the Caribbean and into the Atlantic.

The U.S. military already enjoys such strategic convenience in several nearby islands. A U.S. Air Force plane was seen stopping over at Barbados' Grantley Adams International Airport this week. Knowledgeable sources said U.S. submarine monitoring equipment for Soviet-Cuban traffic was moved from Barbados to St. Lucia three years ago when Barbados tried to raise the rent.

Ultimately, diplomatic sources pointed out, it is Bishop's relations with Cuba and the Soviet Union that will determine the role of Grenada and its airport in the competition for strategic advantage between the United States and the Soviet Union. "The complexion of the government is the real question," said one. "The airport can be used for anything they want."

Alluding to this, Bishop warned Reagan in his second letter that continued U.S. hostility would be likely to push Grenada closer to Cuba and the Soviet Union.

"If you should choose to allow this letter also to go unanswered as happened to my earlier letter to you of March 26, 1981, referring to similar unhappy developments and requesting high-level discussions, then we shall have to conclude that your government does not desire even normal and minimum relations with my government, in which event we would be obliged to consider further measures necessary to advance, consolidate and defend the social, political and economic transformation process which we have undertaken in Grenada," he wrote.

Deputy Prime Minister Coard, asked to declare flatly that the Soviets would not be allowed to use the new airport for military flights, declined, adding: "To threaten a small country and to try to bully it is a very strange way to protect U.S. interests, and it is calculated to bring about the opposite results."

Some Grenadan actions also seem calculated to generate maximum concern among U.S. officials. When a Cuban ship pulled into St. George's harbor one night last August, for example, all the capital's lights went out and soldiers manning roadblocks prevented access to the area while heavy trucks ground away from the docks with concealed loads.

One of those inconvenienced was Bishop's mother, who according to neighbors had gone to visit a bereaved relative and was stopped by soldiers on her way home.

The mystery gave rise to various lines of speculation, including one by U.S. officials that Mig aircraft in crates might have been unloaded. Diplomats now say, however, that the ship brought in six BTR60 armored personnel carriers and a few antiaircraft guns--all of which were driven on display during March 13 celebrations of the 1979 revolution that brought Bishop to power.