If this is Solomon Grundy at his most powerful (not like the last incarnation that was far smarter than his normal level), who has been shown close to Superman's level of strength and the ability to stand toe to toe with Superman, taking his best blows, then I don't think that this battle would be over quickly.
These more powerful versions of Grundy would I think lose to Doomsday but would be able to match him blow for blow fro quite a while.
If it was the latest version of Grundy who looked a lot weaker and less durable, then it would be a short battle, with Doomsday destroying him.
It would be very long, countinuous, and epic (TWSS) fight. But in the end it would be Doomsday. If one killed the other, they would come back, BUT Doomsday comes back immune to what killed him before. After a while, Grundy would just run out of options. I think half, if not the more, of the world would be in ruins by then.
Didn't realize how long ago this thread died. Oh well, f--- it. There it is.