Martinez has returned to his home in the D.C. area and does not need any more procedures, Rizzo said.
The GM said he expected doctors to clear Martinez to travel but wouldn't speculate when that happens.
Rizzo had told reporters Monday that Martinez had a minor cardiac catheterization that day in Washington.
Martinez, who turns 55 on Sept. 26, experienced chest pains during Washington's home game Sunday against Atlanta, left in the sixth inning and was taken to a hospital.
Bench coach Chip Hale will manage the Nats during Martinez's absence.
First-base coach Tim Bogar will be in the dugout and assistant hitting coach Joe Dillon will replace Bogar at first.
The Nationals saw their top National League wild-card lead shrink to a half-game over the Chicago Cubs after Monday night's loss.
Martinez is in his second season managing the Nationals. After winning 82 games last season, the Nats have matched that total this season with 13 games to play.
Martinez had a 16-year MLB career with nine teams.