The White House may have erred by soliciting the opinion of Virginia Sen. Harry F. Byrd Jr. on a proposal to use independent commissions in the selection of federal judges, according to the chief of President Carter's congressional liaison office.

Frank Moore, head of the liaison office, confirmed that the letter, sent by President Carter to all members of the Democratic Caucus, had prompted angry protests within the Virginia Democratic Party.

Using the letter, Byrd, who left the state party seven years ago to become the Senate's only independent, announced last week that he will play a role in the selection of federal judges in Virginia. That is a political prerogative that many Virginia Democrats thought he had lost when he left the party.

The issue has become increasingly important to many Virginia Democrats because legislation is pending in the Senate that would create four new U.S. District Court judgeships in the state. Byrd said last week he was naming two commissions to advise him on who should be named to the positions and said he was doing so at President Carter's urging.

Moore said Monday that the letter to Byrd only asked for his opinion on the use of such commissions and said it was mailed from his White House office late one night. "I probably would have pulled it, if I'd been there," he said.

Since Byrd is an independent who sits with the Democrat caucus, White House and Justice Department officials have said they are uncertain how he should be treated in the selection of judges in the state. "He's neither fish nor fowl," Moore said.

Virginia Democrats ran retired Navy Adm. Elmo r. Zumwalt against Byrd last year and Carter, speaking in Virginia during his campaign, supported Zumwalt. Rep. Herbert: E. Harris (D-Va.) was trying to develop a proposal for a Virginia judicial commission that would have reported directly to President Carter when his proposal was undercut by appointment of the two Byrd commissions.

Despite the furor among some Virginia Democrats, State Democratic Party Chairman Joseph T. Fitzpatrick has said that he believes Byrd can properly claim a role in advising Carter on judges. That issue, he said, was settled in January when Senate Democrats refused to consider Virginia party complaints about seating Byrd in the caucus.