Three Lutheran churches voted overwhelmingly yesterday to merge into a single denomination of 5.5 million members, forming the nation's third-largest Protestant denomination, thereby ending two centuries of separation along ethnic lines.

Meeting separately in Louisville, San Diego and Cleveland, the 3 million-member Lutheran Church in America, the 2.4 million-member American Lutheran Church, and the 108,000-member Association of Evangelical Lutheran Churches approved the merger in secret balloting. The results were announced in a three-way teleconference hookup at 5 p.m.

A roar of approval went up in the three cities as the Rev. Dr. William H. Kohn, head of the AELC, reported: "136 delegates voting: 136 'yes,' no 'no' votes." But there was even greater elation when the Rev. Dr. David W. Preus, whose slightly more conservative ALC convention in San Diego had earlier registered some reservations, reported an 897-to-90 vote of approval. The LCA vote was 669 to 11. Formation of the merged church is set for 1988.

The only major Lutheran body remaining outside the merger plan is the 2.6 million-member Lutheran Church-Missouri Synod, a deeply conservative body whose theological quarrels of the last decade led to the exodus of moderates who four years ago formed themselves into the AELC church as a holding action in anticipation of the merger approved yesterday.

The three merging churches share a strong commitment to ecumenical cooperation and to social action. On the agenda of each convention, for example, is a resolution urging church support for a nuclear freeze.

Each church looks to the Bible as the foundation of faith; each insists on understanding the Bible in terms of modern scholarship rather than interpreting it literally.

The first Lutheran church in this country was founded by Dutch settlers in New York in the mid-17th century. But it was the substantial numbers of German immigrants in the 18th century -- settling in eastern Pennsylvania, the Piedmont region of the Carolinas and the Hudson and Mohawk Valleys of New York -- that established Lutheranism here.

Succeeding waves of immigrants from Sweden, Denmark, Norway, Switzerland and Finland, bringing their Bibles, feather beds and farming tools, settled in the Midwest. With them came their pastors, to establish, in the new world, churches with the familiar rituals and language of the old.

As the immigrants became Americanized, and German-Americans mixed with neighbors of Swedish or Finnish extraction in public schools, the Grange and Rotary meetings, separate ethnic churches became anachronistic.

So, in the early 1960s, when the ecumenical mood was strong in Protestantism, the consolidation began. In 1960, three Lutheran denominations -- German, Danish and Norwegian -- in the Midwest united to form the American Lutheran Church, with headquarters in Minneapolis. A fourth, the Lutheran Free Church, joined in 1963.

In 1962, the Lutheran Church in America, with headquarters in New York City, was formed by a union of the United Lutheran Church (German), the Augustana Lutheran Church (Swedish), the American Evangelical Lutheran Church (Danish) and the Suomi Synod (Finnish).

Secular trends of the last two decades -- the civil rights and communications revolutions, increased urbanization and greater mobility of Americans -- further blurred the ethnic distinctions that once formed denominational boundaries.

Today, both the ALC and LCA churches, with their Scandinavian and Teutonic roots, have strong black caucuses. Both churches have taken strong stands and national leadership positions on civil rights and peace issues.

The proposed new church will be surpassed in size in American Protestantism only by the 13.6 million-member Southern Baptist Convention and 9.6 million-member United Methodist Church.

Yesterday's Lutheran merger vote follows a similar decision earlier this summer by the United Presbyterian Church and the Presbyterian Church in the United States to heal their rift dating back to the Civil War. The Presbyterian action remains subject to approval by regional bodies.