A Rem 700 stock will not fit a Rem 600, they are different sizes. Jsut becasue you have a short action does not mean another short action stock will fit. FWIW, the Rem 600 is the same basic action as a Rem Model 7 with some minor differences, so even a Model 7 stock will not fit without some modification. I would recommend either buying a replacement stock made for your rifle (it will be a chore to find one, and it won't be cheap), refinish the stock you have in a way that will be more durable (Remington varnished those stocks, so they look like hell once they start to chip and wear), or just hunt with it as it is.
I do stock work for a living, and often refinish stocks using Brownells' Acra-Wood epoxy finish. It is as tough as nails, but it takes some care to apply it well.