The Cook County Democratic organization today finally voted to give its formal and unanimous endorsement to the party's nominee for mayor, Rep. Harold Washington (D-Ill.).

But about a third of the committee's members--most from ethnic neighborhoods where party voting strength is strong and support for Washington is low--passed up the meeting, signaling a still elusive unity in this overwhelmingly Democratic town as the April 12 mayoral election nears.

Washington, whose maverick campaign and repeated denunciations of coveted party patronage have alienated many Democratic regulars, welcomed the endorsement as a sign that the party was uniting after the bitter, racially tinged three-way primary Feb. 22.

In a 10-minute acceptance speech, Washington also offered a gesture of conciliation to critics.

"I've never been accused of having a concrete head--maybe rubber, but never concrete," Washington said. "I can be penetrated by reason, by logic, to a certain extent by tradition and the necessities of the time. So we will have an open door and it will stay open."

Still, Cook County Democratic Chairman Edward R. Vrdolyak conceded after the half-hour session at the Bismarck Hotel near City Hall that the party was still not solidly behind Washington, who is bidding to become the city's first black mayor.

"It's no different than what we went through in the primary," Vrdolyak said. "The organization was not 100 percent together. It is not 100 percent together now." But Vrdolyak said he still thought the party could win the general election.

A few ward committee chairmen--the party bosses--have endorsed Republican nominee Bernard E. Epton, and more are expected to follow suit in the wake of Mayor Jane M. Byrne's decision Wednesday to abandon her write-in campaign after just one week.

Byrne, who finished second to Washington in the primary, dropped her candidacy when the city's elections board refused her request to simplify Chicago's awkward write-in procedures.

Theoretically, the organization's endorsement should bring Washington some money, and, most important, the armies of precinct captains and other workers under the command of the ward bosses.

Yet Vrdolyak declined to answer when asked if even those who voted today to endorse Washington would enthusiastically support him with their organizations. The endorsement vote was vocal, sparing members the potential embarrassment of a roll call.

Most of those absent were from ethnic neighborhoods in the northwest and southwest sectors, where there is fear of a black mayor who might empower civil rights leaders, be too aggressive at pushing neighborhood integration and be insensitive to the concerns of whites, who constitute 60 percent of the city's 3 million people.

The meeting was marked by strong appeals from some longtime black ward bosses that their white colleagues help blacks enter the city's melting pot of political power.

"For many years, our people have been here working in the vineyards of the Democratic Party," said Cook County Commissioner John H. Stroger Jr. of the South Side. "I guess a lot of you thought we were the coal to heat that pot while you melt. Now we want to get in that pot and melt with you and we need your help."

The local party has come under increasing pressure from national Democrats to close ranks behind Washington both as a sign that it will treat blacks fairly and to tune it up for the 1984 presidential election.

"The whole country is looking to Chicago with a high, with tremendously great expectations," Washington said. "They want us to come together and forget past frictions and fractions and animosities . . . ."