The nation's unemployment rate inched up to 7.8 percent in July, the same rate as in May and only one-tenth of a point higher than in June, the Labor Department reported yesterday.

"We are moving on a plateau," Treasury Secretary G. William Miller said. "We have had three months without any significant increase in unemployment."

Miller told the congressional Joint Economic Committee that the Carter administration believes the recession, which began in January, is "bottoming out" but that a recovery has not yet begun. He reiterated the administration's opposition to congressional action on a tax cut before the November election but added, "We see the recovery as being too sluggish if we do nothing."

A tax cut focused on spurring business investment and on offsetting part of the increase in Social Security payroll taxes set for Jan. 1 may be appropriate in 1981, Miller stated as he has numerous times in recent weeks.

As evidence the recession is nearing its low point, the Treasury secretary cited the 2 1/2 percent increase reported this week in the index of leading indicators for June, the large 30 percent jump in housing starts for the same month, an incrase in June retail sales even after adjustment for inflation, a pickup in auto sales in the first part of July and the latest employment data. p

Separately yesterday, the Commerce Department said construction spending fell 2.3 percent between May and June to a seasonally adjusted annual rate of $215 billion, which also was 4.7 percent below the June 1979 rate.

Janet L. Norwood, commissioner of labor statistics, echoed Miller's statement about a "plateau." "The July . . . figures suggest some leveling off in the labor market indicators," she told the committee. "Total employment, as measured by the household survey, after declining steadily since February, actually rose by 460,000."

The number of persons unemployed rose to 8.2 million from 8 million the month before. The size of the civilian labor force increased by a substantial 661,000 persons to a total of 105.2 million.

The unemployment rate for adult men was unchanged in July at 6.7 percent. However, Norwood noted, "Adult men have borne the brunt of the increase in joblessness since the recession began; their jobless ranks have swelled by over 1 million persons," accounting for about two-thirds of the total increase in unemployment. In January, the unemployment rate for adult men was 4.7 percent.

The jobless rate for adult women rose from 6.5 percent in June to 6.7 percent in July. Except during recessions, the unemployment rate for women is usually higher than that for men.

The jobless rate for teenagers climbed from 18.5 percent to 19 percent.

Unemployment among white persons increased from 6.8 percent to 6.9 percent. The rate among blacks climbed more, from 10.2 percent to 10.9 percent. For the 387,000 black and other nonwhite teenagers, the rate rose from 34.4 percent to 36.6 percent.

The length to the average work week declined from 35.1 hours to 35.0. In manufacturing -- where the bulk of the increase in unemployment has occurred -- the work week was unchanged at 39.1 hours, while the average worker was putting in 2.5 hours of overtime.