Country singer Johnny Cash is doing well in a Nashville hospital after surgery to repair scar tissue and adhesions from the 1983 removal of part of his stomach because of ulcers. He is expected to spend about a week in the hospital . . .

If you thought you saw Clint Eastwood around the Madison Hotel recently, you did. The stony tough-guy actor stayed there while he was in town for yesterday's presentation of the National Medal of Arts Awards at the White House . . .

Sen. Edward Kennedy's son Edward Jr. was scheduled to arrive in San Diego yesterday to be fitted with a new artificial leg. The 23-year-old, who lost a leg to cancer when he was 12, chose wood rather than a plastic or rubber leg and asked that the wood be stained the same color as the teak on the family yacht . . .

Washington artist Frederick Hart, who created the statue of three soldiers that stands near the Vietnam Veterans' Memorial, has been appointed by President Reagan as a member of the Commission on Fine Arts for a four-year term . . .

Henry J. Fortunato, who has been senior editor of Regardie's magazine for five years, has told the magazine's owner, William A. Regardie, that he will be going into business for himself. Regardie, who is executive editor of the magazine, yesterday described Fortunato, a Georgetown University graduate, as one of his best and brightest. "It's a terrible loss for the magazine and it means I'm going to have to work harder as editor. I'm learning from the big boys like U.S. News owner Mort Zuckerman and plan to now make myself editor in chief of my magazine." . . .