Western Europe's Socialists emerged, by a small margin, as the most important alignment of parties in the first popularly elected European Parliament, but with far fewer seats than had been at first anticipated.

A West German computer projection based on early returns gave the Socialists 108 of the 410 seats in the nine-nation Parliament. They were trailed closely by the Christian Democrats with 100 seats and the Conservatives with 63. The Communists got 44, according to the projection.

The most striking thing of all, however, was the low turnout in most of the nine countries - an average of 60 percent in a group of states where voter participation is ordinarily much higher.

The turnout ranged from about 30 percent in Britain to 85.9 percent in Italy where voting is semi-obligatory. In Belgium, where the voting is mandatory and nonvoters must pay a fine of about $12, the turnout was about 75 percent, much less than usual.

Italy is, with West Germany, traditionally the most "European" of the nine member countries, while Britain has been the least so. The European Economic Community's 10th member, Greece, does not become a full member until 1981 and did not take part in the elections.

Except for France and Denmark, where the elections were turned into major domestic political debates, this first public campaign for the transnational assembly seemed to come across as an abstract discussion about something that has little direct relationship to people's daily lives.

In France, one of the two nationwide state television networks hardly bothered to report the results in the other eight countries of Europe, concerating on the relative standing of the national parties.

This was, along with the Danish results, perhaps the most significant harbinger for national politics in the Common Market. The French parties are gearing up for the presidential election of 1981.

The big losers in France were the Gaullits, who slipped from 22.6 percent in the March 1978 legislative elections to 16 percent, according to two separate computer projections. The big winner was the slate backing President Valery Giscard d'Estaing, headed by Health Minister Simone Veil. It got more than 27 percent of the vote, compared to 21.4 percent in 1978.

The French Communists slipped from 20.5 percent to 19, and the Socialists almost held the 24.7 percent they won in 1978. The French voter turnout was 60 percent, compared to 83.4 percent in 1978.

In Denmark,where the only major openly anti-European slate was campaigning, the country's leading party, the Social Democratic, was given a stinging rebuke by the Voters, gaining only three of Denmark's 16 seats in the Parliament.

The anti-European party got five seats and other nonsocialists seven seats, with no results in for the seat in Greenland, Denmark's arctic colony.

The poor Socialists showing in Denmark and elsewhere apparently is related to the low turnout. Socialist voters seemed to be disproportionately represented among the nonvoters.

In Denmark, only 49 percent of the voters turned out, compared to 89 percent in the 1977 Danish national elections.

In Britain, the Socialists also did poorly. One computer projection based on partial returns had the Labor Party slipping from the 37 percent they scored in the British general elections last month to less than 28 percent.

The Conservatives were expected to win 43 percent, almost one point less than last month, but more than enough to sweep almost 60 of Britain's 81 seats.

Britain was the only country where the candidates ran in single-member constituencies. Elsewhere, they ran on nationwide or regional slates.

Environmentalist groups did particularly well throughout Europe. In France, they got 4 percent, the highest vote after the four main French parties.

West Germany's ruling Social Democratic Party suffered from the general European-wide trend against the Socialists, slipping from 42.6 percent in the last national election to about 41 percent today.

The German Christian Democrats and Christian Socialists rose from 48.6 percent to 49.5 percent, and were projected to win 43 of West Germany's 81 seats in the European Parliament in Strasbourg. The Free Democrats, West Germany's Liberal Party and the junior allies in the ruling coalition, slipped from 7.9 percent to 6.1.

Former West German chancellor Willy Brandt, who would be considered a leading candidate for the presidency of Europe's first elected assembly, if not for his failing health, expressed disappointment over the Socialist showing. He also deplored the low turnout of about 60 percent in West Germany.

In Italy, the trend in national elections a week ago against the two major parties, the Christian Democrats and the Communists, was confirmed with even poorer showings for them. The Communists slipped further, from 9.8 percent to 11.4. This, howtian Democrats went from 38.3 last week to 36 percent today.

Bucking the general European trend, the Italian Socialists rose from 9.8 percent to 11.4 This, however, simply may reflect the emerging pattern in Italy of scattering the vote among a hose of minor parties. Ten different Italian parties are expected to be represented in Strasbourg among Italy's 81 European deputies.

The Italian Radical Party did even better than last week, moving from 3.4 percent to 4 percent.

In Belgium, both Socialists and the Christian Democrats gained at the expense of smaller parties. The Christian Democrats were expected to go from 36.3 percent in the last national elections to 49.5 percent today. The Socialists were expected to rise from 25.4 percent to 41 percent.

No results were available from the Netherlands, where counting is not scheduled to start until Monday. A poll of Dutch voters as they left the polling places showed the Socialists down slightly and the Christian Democrats up.

The Dutch voted on Thursday, and the ballot boxes were sealed.

The British, Irish and Danes also voted on Thursday, but their ballot boxes were also locked up pending the voting in the other five countries today. No results were announced from Ireland today either.

In Luxembourg, the community's smallest country with a population of 360,000 the coalition government of Prime Minister Gaston Thorn, a Liberal, was defeated in national elections that coincide with the European vote. The Thorn government, which fell largely because his Socialist coalition partners lost ground, is expected to be replaced by one headed by Social Christian leader Pierre Werner.

Thorn was widely expected to lose and to present his candidacy as president of the European Parliament.

Apart from being unable to see how the assembly with its limited powers can deal with basic problems, many Europeans apparently stayed away from the polls because they failed to understand that electing the European Parliament by popular vote almost inevitably will increase its power and importance.

The European Parliament has existed since the Common Market was founded in 1957, but its members were appointed by national parliaments from among their own members. The appointed Parliament had only 198 members.

Its deputies had broad theoretical powers over the budget and over the Executive Commission of the European Community in Brussels. But their lack of a popular mandate served as a strong inhibition against exercising those powers.

When the new, elected European Parliament first meets July 17, it is expected to begin flexing its new-found muscle rapidly. Barbara Castle, who led the British Labor Party slate, was asked after her election how the relatively weak body could change things.

"You don't need to be able to make laws in order to make a lot of noise," she replied. CAPTION: Picture, French President Valery Giscard d'Estaing goes to the polls in Chanonat, France. AP