Yup...you can do that. But it's cooking the soup with the ham bone that makes it best. Without it, the peas have that grainy, mealy texture.
So what I often do, after a ham dinner, I cut most of the leftover ham off the bone.....throw it (the ham bone) in the soup pot with some suateed onion and the split peas and water.....and let it simmer all day. Put it in the fridge overnight and then in the morning, skim any fat off that has risen to the top and hardened.
THEN add your veggies, and cubed ham....let it simmer again for several hours (or until your veggies are al dente. Pea perfection.
*another trick for great homemade soups, is I always throw in a handful of the dried 'soup mix'. You can get it in some stores in the bulk department. It only contains dried veggies.....leeks, carrots, green stuff...lol....it has no seasonings.