A private corporation that already runs one hospital here has signed a contract with the Navy to take over the $22 million F. Edward Herbert Hospital, which has been the target of investigations since its opening two years ago.

The leasing of the cost-overrunn plagued hospital to Westbank Medical Center Ltd. occured despite an attempt to stop it made by the rarely contradicted Louisiana corporation that makes recommendations on which hospitals should receive federal funds.

Jessie Smallwood, executive director of the New Orleans Area-Bayou-River Health Systems Agency Inc., last week called on President Carter to keep the Navy from trying to "dump its white elephant." She had recommended closing it because, she said, the hospital, which among other things could have utility bills of as much as $92,000 a month, would be too expensive for anyone to operate.

But the contract was signed, about a month before the completion of a $20,000 study on alternative uses for the hospital. The facility was supposed to be part of a massive military expansions here that never materialized. Since the hospital's opening, a government study showed that the average number of patients in the 250-bed facility on any day was 23.

Smallwood said that there are already 2,500 too many hospital beds in the New Orleans area. However, the Westbank spokesman said, there are not enough beds on the west bank of the Mississippi River, where that hospital is.

Because of this shortage of beds and because it will operate such services as an obstetrical-gynecological clinic, a nursing area fo for elderly patients who require specialized care, areas for physical rehabilitation and alcoholic rehabilitation and, possibly, a hospice-a unit to prepare terminally ill patients for death with dignity-corporate officials are convinced that they can operate the hospital profitably, the spokesman said.

In Westbank's lease, it guarantees the Navy $480,000 per year, but, the corporation spokesman said, that amount will rise if hospital income increases.

The corporation will not change the facility's name, said the Westbank representative.

The hospital had been named for the man who represented Louisiana's First Congressional District for 36 years and had planned for the military buildup when he was chairman of the House Armed Services Committee. However, that idea died when he was removed from control of the committee in January 1975.