Washington — With virtually no public opposition, President Reagan is moving to establish full US diplomatic relations with the Vatican. On his instructions, the State Department is now exploring the subject with Roman Catholic Church officials in Rome. Once the talks are completed, and if formal relations are established, say administration officials, Mr. Reagan will then nominate an ambassador. There have been reports that he intends to name William A. Wilson, a California land developer who is now the President's personal envoy to the Holy See.
The development comes after the President's signature of a measure repealing an 1867 law prohibiting the expenditure of federal funds for a diplomatic mission at the Vatican. The legislation was in the form of an amendment to the State Department authorization bill sponsored by Sen. Richard G. Lugar (R) of Indiana and supported by the President.
Administration officials tend to stress the foreign policy benefits of formal ties with the Holy See, including greater access to the Vatican's worldwide channels of information. But Mr. Reagan is also conscious of the domestic political benefits as he courts the vote of Roman Catholics and organized labor.
Religious observers say a marked change in the American climate in recent years accounts for the fact that the issue of Vatican relations sailed through the Congress with scarcely any media attention. A few Protestant organizations, including the National Council of Churches and the Baptist Joint Committee, opposed the measure at every legislative step. But they were unable to arouse significant public interest. Some 20 years ago President Truman was unable to appoint an envoy to the Vatican because of public opposition.
''The focus of people has changed,'' comments Dean Kelley, director for religious liberty at the National Council of Churches. ''Where religious liberty is concerned, the perceived danger 30 years ago was an expansionist Roman Catholic Church. Today our view of the church is less apprehension-ridden, and the perception is that the danger is an expansionist US government - IRS scrutiny of mail-order ministries, for instance, or efforts to clamp down on the fund-raising tactics of some groups.
''But it is inappropriate for the government to send an ambassador to a church anywhere, anyplace,'' Mr. Kelley adds.
Other Protestant observers cite the ecumenical movement, a reluctance to appear bigoted, the religiously ''correct'' presidency of John Kennedy, and the world stature of Pope John Paul II as further factors in the relative absence of public concern about the establishment of formal Vatican relations.
Believing that such relations would violate the constitutional principle of separation of church and state, organizations like the Americans United for Separation of Church and State (AU) will voice their opposition to nomination of an envoy to the Holy See. But AU officials say it is not clear whether the issue could be litigated. ''We are in a murky area of law when we get into foreign policy,'' an AU official says. ''We will take a strong stand in the Senate confirmation hearings, but we are still studying the legal implications.''
The issue is not one which Roman Catholics have been interested in, according to Catholic sources, and would become so only if suddenly there were a strong public reaction against the presidential move. But officials of the National Conference of Catholic Bishops say Catholics are gratified by this ''reasonable, overdue step'' as the original 1867 law was in part a response to a wave of anti-Catholicism in the United States.
''A lot of opposition since then has reflected an anti-Catholic spirit,'' says Russell Shaw, director of public information of the Bishops' Conference. ''So it's great that the administration, the Congress, and the country have reached a level of maturity and decency that have gone beyond that kind of bigotry.''
Some Catholic leaders, however, voice concern that the move may stir up more interreligious unhappiness. They also fear that the Vatican's appointment of a nuncio (official delegate) or pro-nuncio accredited to the US government might reduce the American church's independence of action. (At present the Vatican has an apostolic delegate to the church.)
The State Department supports the establishment of full diplomatic relations, citing not only the advantage of broadening channels of information throughout the world but of showing public approval for a Pope who is using his influence in what is regarded as a constructive way.
''This Pope especially has been very brave in taking stands with which we agree,'' said one US official. ''Also, many other nations of all political spectrums have recognized the Vatican. We would be joining our major allies.''