Do eyeballs stay the same size for life?

No. No, that would be one of those "medical myths." eyeballs grow, and the shape of the eyeball can change through the growing years.
No. Eyeballs grow with your body. In addition, eyeballs can change shape over time, which can result in vision changes (some people's eyeballs grow longer than wider which changes the ability of the lens to focus visual images on the retina).
No. Eyes grow as your baby grows, just like his/her other organs. In addition the iris, the colored part of the eye, likely will change color during the first year. Most babies are born with blue eyes, but watch closely as your baby's eyes get more color and depth to them!
No. Eyeballs grow in diameter a very small amount each year during childhood, until they reach the adult diameter of 1 inch.