A wide public opinion gulf separates the United States from much of Western Europe on the need now for massive defense spending increases and the advisability of renewed arms limitation negotiations with the Soviet Union.

As President Reagan appears ready to squeeze social programs to help balloon the U.S. defense budget, key West European politicians say they have no mandate from their voters to do the same, and consequently show reserve toward the sharper calls from Washington for boosts in their own military efforts.

Trying to bridge this Atlantic gap, U.S. officials now speak of the Reagan administration's intention to "lead by example" -- to demonstrate first that the United States is determined to do what is necessary to fund defense even though doing so hurts, and then to urge that the European allies do likewise.

If European public opinion happens to contrast with that in America, Reagan officials suggest that European opinion should be changed -- through open discussion and public education campaigns underlining the more menacing Soviet threat today not only toward Western Europe but around the world, especially in view of Western military deficiencies that have been allowed to develop.

This message is just beginning to roll through West European capitals, and the initial European response predictably has been cool.

Having welcomed Reagan's election last autumn for the consistency and coherence that it was hoped for new president would bring to U.S. foreign policy, West European officials now are concerned about the potential spillover to this continent of Washington's tougher stance toward Moscow and heavy emphasis on military spending four weeks after the inauguration.

Pointing to their own hard-pressed economies, which individually are much less shock-resistant than America's -- although together they exceed the gross national product of the United States -- the West Europeans claim to have less economic maneuvering room than Washington to increase defense expenditures.

While saying they are prepared to take up a greater share of the existing defense burden in the European theater, the West Europeans would prefer to do this by improving efficiency -- that is, by getting more for their defense marks or pounds rather than by spending more of them.

On top of this, answering the Reagan administration's call for a more realistic approach to East-West relations, the European allies say they had never been naive, and the Bonn government in particular continues to argue that cooperative ties with the Soviet Union are essential for stability in Europe.

Regarding one of the most central strategic questions for the Western alliance -- namely, how to restore the East-West military balance in Europe -- the Reagan administration contends that a buildup of arms in the West is first necessary in order to gain bargaining leverage over the Soviets, while the West Europeans, doubtful about ever catching up with Moscow's military edge in the European theater, urge that renewed arms limitation talks get under way immediately.

These differences in political priorities and public mood were highlighted during a weekend of honest discussion between high-level officials and defense experts from a number of Atlantic alliance countries. This annual forum, sponsored by a private group in Muncich, has served traditionally as a bellwether for the themes and tones to come in alliance relations.

With the U.S. case stated forcefully there by Deputy Defense Secretary Frank Carlucci, Sen. John Tower (R-Tex.) and others, the impression was left on European participants of an America that now has a definite idea of where it wants to go.

The challenge put to Europe was to get alongside the United States. But the worry voiced by some West Europeans is that the United States will get too far ahead and then pull hard for Europe to catch up, failing to appreciate the contrast in political demands and economic strengths.

More wear and tear in the U.S.-Eurpean relationhip seems unavoidable. But so does some additional give and take. The inter-alliance strain of the past year did, after all, also produce some convergence.

On the other side, Western Europe has been nudged into considering a more coordinated -- if still informal -- effort to extend the military, diplomatic and economic bulwarks of the North Atlantic Treaty Organization beyond Europe to strategic zones in order to protect sources of important minerals and oil and to counter instability in the Third World.

In turn, Washington officials readily acknowledge the need for improved consultation and flexibility in dealing with the Europeans, and have adopted the concept of a "division of labor" within the alliance that was pressed on Washington by West German officials following the Soviet invasion of Afghanistan.

To bring Europe into step with Reagan's priorities will be more formidable task, however, requiring more sensitivity to European political sensibilities than the new U.S. administration has shown and a greater willingness to sacrifice than the West Europeans have volunteered.

One thing made clear to the U.S. officials who came to Munich was that it no longer is enough -- in fact, it may be counterproductive -- for a West European politician to ask his country to do something because it would be pro-American. "We must tell our voters what we do is in our interest," declared Horst Ehmke, a leading Social Democratic member of the West German Bundestag.

Ehmke said further that how America behaves towards Latin America would critically affect European opinion, warning that a U.S. military intervention would bring a wave of anti-Americanism particularly among European youth.

For West Germany's part, Defense Minister Hans Apel did at least promise that the Bonn government would move to counter growing pacifist sentiment in the country by putting the case for a strong Western defense more strongly to the West German public.

The European contrast with the United States also differs in degree, depending on the specific country involved. In France, the public is more inclined toward an increased defense budget. In Britain, there is not the same interest in detente as there is in West Germany, where perhaps the public mood is most sharply at odds with the American attitude.

But as West German Gen. Gerd Schmuckle told Tower, "Naturally, senator, the Europeans are complicated allies. Only where in the world would you find better?