My goldfish begs for food, so I give him a bit of lettuce every once in a while, am I doing something wrong?

I don't know about the egg whites but its good for goldfish to give them some vegis occasionally. Also, a 15 gallon isn't ridiculously large its actually going to be ridiculously small soon. If you have a feeder goldfish it will grow to over 12 inches. A fancy goldfish can grow to 8-10 inches.