Congressional Republicans fell into the awkward position yesterday of fighting among themselves over a Republican appointed to a job by a Democratic President.

Senate Minority Leader Howard H. Baker Jr. (R-Tenn.) and House Minority Leader John J. Rhodes (R-Ariz.) led the onslaught against the appointment of Republican Samuel D. Zagoria to a GOP seat on the Federal Election Commission.

President Carter, they charged in separate statements to the Senate Rules Committee, had reneged on a promise he made them Feb. 23 to make the appointment from names they submitted to him.

"Republicans have been had, and I, for one, intend to do everything in my power to see that it does not happen again," Baker said.

"We are confronted with the prospect of having the President who is leader of the majority party, make the determination of who shall be a very important spokesman and advocate for the minority party. It believes all concepts of fair play," Rhodes said. "This would be a long step toward imposition of one-party rule in this country."

Furthermore, Rhodes said he doubted Zagoria's credentials as a Republican and sees "nothing in his record that indicates a familiarity with federal campaign law."

But Zagoria, director of the labor-management relations for the U.S. Conference of Mayors, had at least three liberal Republicans squarely in his corner.

Sens. Clifford P. Case (R-N.J.) and Charles McC. Mathias Jr. (R-Md.) lavishly praised Zagoria's abilities as a congressional aide and political operative, with Mathias saying, "I can testify from experience that he is a Republican interested in the party." And former Kentucky Republican Sen. John Sherman Cooper wrote a letter to the committee supporting the nomination.

The Rules Committee adjourned without acting. Observing the conflict between Republicans, committee charman Howard D. Cannon (D-Nev.) said bemusedly, "This is their fight, I hope they can get their act together."

Zagoria, 58, was nominated last month for a six-year term on the FEC, a $50,000-a-year post. He was a reporter with The Washington Post before becoming administrative assistant to Case. He was a member of the National Labor Relations Board from 1965 to 1969.

He said yesterday that he has no intention of asking that his name be withdrawn.