 Because of the natural acidity in ketchup, yes, it is shelf-stable, but with a caveat. The stability after opening it can be affected by storage conditions, which means if your kid opens the refrigerator 18 million times, that ketchup at some point will oxidize. And while it won't go bad, Heinz says, the color will change, the flavor will be lost. And so you're better off just refrigerating after opening.