The nitric acid forms in the outlet of the ozone generator, and will not get into the tub.
It forms in very small quantities, and usualy dries out and stays in the tube in the form of a yellowish coating or gel.
It can be cleaned out with water, being careful not to get it on broken skin or in your eyes.
If a small amount of it got into the tub, it would lower the pH slightly, but otherwise go unoticed.