The Cuyahoga County Democratic Party today became the first big city organization in the country to turn against President Carter and urge Sen. Edward M. Kennedy (D-Mass.) to challenge for the party's presidential nomination.

Party leaders met tonight and voted 16 to 4 to schedule a county convention of precinct committee members for May 26 to draft Kennedy for president.

Timothy F. Hagan, the county chairman, said Democrats here have a philosophical difference with the direction Carter is taking the party and the country.

"Democrats traditionally speak for the poor and the powerless," Hagan said. "We are not going to let the Carter administration redefine the purpose of the Democratic Party without a fight."

Hagan reminded a reporter that 20 years ago party then-chairman Ray T. Miller became the first big city leader to back John F. Kennedy for the presidency. When it was pointed out that the White House occupant at the time was a Republican, Dwight D. Eisenhower, Hagan replied, "I don't see much difference between then and now."

Though Hagan has heard nothing from the White House, he has received calls from state party leaders and Kennedy aides Rick Burke, who have asked him to call off the movement. Hagan said he refused the requests. "If enough people in the Democtatic Party urge the senator to run, he'd have to give it consideration," he said.

"The question is," Hagan said, "would he be splitting the party? I don't think he would. This is a grass-roots movement."

Another Ohioan, state Sen. Jay Timothy McCormack, this week announced plans for a statewide Kennedy-for-President Committee, but state party leaders remain loyal to Carter.

Practically speaking, a serious revolt by Cleveland Democrats would weaken Carter's reelection chances. It is generally agreed that any Democrat must pile up a huge majority in Cuyahoga County in order to carry Ohio, something the president barely did in 1976.

Hagan's actions have put him in the unusual position of being allied with Cleveland Mayor Dennis J. Kucinich. Hagan and the regular Demovrats have made defeating the maverick Kucinich in the mayoral election this year their top priority.

But in a recent Playboy magazine interview, Kucinich sharply critized Carter and California Gov. Edmund G. (Jerry) Brown Jr., and had kind words for Kennedy.

Hagan said this was a good indication of Kennedy's ability to bring together the disparate elements of the party.