Just how much effect the U.S. air raid against Libya will have in restraining state-backed terrorism remains to be seen. But the bombs that landed on Tripoli and Benghazi last week have already given new life to an old debate about the relationship between the United States and Western Europe.

This week, newspapers and television stations across the Continent carried stories about the "vanishing Americans."

"Yankee Stays Home," headlined The Economist, the British weekly. "America Flees the Old World," announced the Paris daily Liberation.

From Athens to London, the Stars and Stripes were being hauled down as European-based U.S. institutions sought to avoid becoming the target of a terrorist attack. There were reports of Hollywood figures Sylvester Stallone and Steven Spielberg deciding to give the Cannes film festival a miss this year, American cyclists pulling out of the Tour of Spain, U.S. tour groups diverting from Venice and Paris to Florida and the Caribbean.

The spectacle of a diminishing American presence has, in turn, provoked other questions about the value of the western alliance 40 years after the end of World War II, President Reagan's sense of judgment and the willingness of Europeans to face up to terrorism. With just a week to go before the seven-nation summit of industrialized nations in Tokyo, Western Europe finds itself plunged into a new round of introspection about life in the shadow of the superpowers.

The soul-searching has taken different forms in different countries. In Britain and West Germany, pro-U.S. governments have been strongly attacked by left-wing opposition parties for defending the American action. Here in France, the most vocal protests have come from right-wingers disillusioned by the new conservative government's refusal to grant a U.S. request for overflight rights for F111 strategic bombers based in Britain.

The contrast between the French and British reactions to the air raid is revealing because it captures the paradox of the trans-Atlantic relationship. In Britain, the country that facilitated the Reagan administration's attack on Libya, the U.S. air raid was condemned by two-thirds of the population, according to a Gallup poll. In France, which took care to disassociate itself from the whole business, the same poll showed a 61 percent approval rating for the U.S. action.

At the same time, other polls showed that a majority of French people supported their government's decision not to allow the F111s to fly over France.

The only way of making sense of these apparently contradictory findings is to put them in the broader context of Western Europe's sense of dependence on the United States. Anti-American sentiment tends to be strongest in countries, such as Britain or West Germany, whose foreign policy is perceived as tied to Washington. It is weakest in a country such as France, which has a highly developed sense of its own independent interests.

In France's case, the domestic debate over the U.S. attack on Libya has been complicated by the power-sharing arrangement known as "cohabitation," which arose from the narrow right-wing victory in last month's parliamentary elections. The decision to refuse Washington's request for overflight rights was made jointly by Prime Minister Jacques Chirac and Socialist President Francois Mitterrand, whose mandate does not expire until 1988.

As a result, the conservative prime minister found himself being criticized by some of his political allies, including former president Valery Giscard d'Estaing, but in agreement with Mitterrand, a longtime political enemy. It was an uncomfortable position for a politician who had made the struggle against terrorism in coordination with the other western democracies a major theme of his electoral campaign.

To make matters worse, word leaked out via the United States that Mitterrand might have expressed a willingness to go along with "an all-out attack" on Libya aimed at getting rid of Qaddafi once and for all. According to the U.S. reports, the president's comments came in a private conversation with U.S. special emissary Vernon A. Walters two days before the bombing raid.

A similar account of a conversation with Mitterrand has been provided by the West German politician Franz Josef Strauss. Strauss, who met Mitterrand this week, told reporters that the French president was in favor of "a large military attack" on Libya rather than just a simple "pinprick" operation. Mitterrand has refused to confirm or deny the remarks attributed to him.

The prevailing view among informed French officials and western diplomats here is that, while there has been plenty of loose talk about the desirability of getting rid of Qaddafi, none of it amounts to very much. Tough-sounding remarks about Libya are invariably made in the context of an objection to the Reagan administration's specific plans for dealing with the problem.

The arguments advanced by French officials to justify the refusal to facilitate the U.S. bombing raid last week are likely to remain valid in the event of a bigger operation. They include the threat to eight French hostages in Lebanon, a problem that has "obsessed" Chirac, to use his own word, and the fear of an antiwestern backlash in moderate Arab states.

In Britain, too, there are divisions in the establishment on how to handle Libya. Dissent within the ruling Conservative Party forced Thatcher to climb down from her original contention that it would have been "inconceivable" to refuse to allow the Reagan administration the use of military bases in Britain. Her predecessor as prime minister, Edward Heath, told the House of Commons that he did not believe that "bombing cities" was "the right way to destroy terrorism."

The complicated reactions in Western Europe to the bombing raid were reflected in a cover story in The Economist this week, which noted that dislike of Reagan's policies was not the same thing as a general dislike of America. The magazine pointed out that ordinary Europeans are visiting the United States in record numbers, are addicted to American mass culture and like and respect Americans.

The story was headlined: "We Love You, We Love You Not: Gosh, Isn't Life Confusing," an admirable one-sentence summing-up of the trans-Atlantic relationship as it attempts to surmount its latest crisis.