Robert L. Calhoun, the Alexandria Republican generally believed to be the party's most likely candidate for mayor this spring, backed out of the race yesterday, announcing instead that he will seek reelection to the City Council.

The deadline for Republicans and Democrats to file for the May city elections is Friday and thus far no one has challenged Democratic Mayor Charles E. Beatley, who is seeking his sixth term as mayor.

After Calhoun bowed out of the race, council member Carlyle C. Ring, another Republican lawyer, said that he is considering the race and had recently gathered the required number of signatures to file as a candidate.

"If people don't have a choice for mayor the system doesn't work at its best," Ring said. "Our purpose as a Republican Party is to give the people a choice."

Calhoun, a transportation lawyer, said that he decided not to oppose Beatley after considering the amount of time the part-time, $12,500-a-year position requires. Beatley handily defeated Calhoun in 1982, when the two-term council member became the first Republican to run for the mayor's office in more than a century.

"The people of Alexandria have come to expect a full-time mayor," Calhoun said. "That would not be possible if I were to continue as an attorney."

Beatley, a retired airline pilot who says he spends an estimated 60 hours a week as the city's top elected official, said he welcomed any opposition. "An opponent would be good for the city. It would mean more handshaking and coffees . . . . and bring out more voters, which would help my Democratic council members," he said.

On Sunday, the mayor and all three Democratic council members kicked off a joint reelection campaign. Council members Patricia S. Ticer, Donald C. Casey and Lionel R. Hope said that the united effort would save money and unnecessary squabbling among themselves, should there be a need for a Democratic primary.

If more than six Democratic candidates file for the council by Friday, a primary will be held on March 5. To date, the three incumbents, former council aide Del Pepper, and former congressional candidate Craig Coverdale have filed.

The Republicans also will conduct a March 5 primary if, along with council members Margaret B. Inman and Calhoun, at least four others file for the council.