My dog's paw is deeply cut. Does he need stitches or anesthesia?

I'm sorry that has happened to Rocco. Since its been a few days, stitches are not likely to be the best course of action at this point. Once a wound is open for over about 12 hours, we lose the opportunity for good healthy tissue being there and being good for a successful closure. In the future if something like this happens, take him to the vet the same day. Often, in a laceration such as this, just some sedation and a local anesthetic will be sufficient, rather than full anesthesia. Severe lacerations, however, do often require general anesthesia. If he is otherwise healthy, there is a greater risk in complications of healing of a wound not adequately cared for than there is a risk of the sedation or anesthesia.
Take him to your vet today if at all possible. They can evaluate it, determine whether or not stitches are an option at this point (again, I think its probably not advisable but I would defer that to the veterinarian who examines it today). They will likely also prescribe antibiotics so that infection does not become a problem.
Keep a Collar (cone of shame) on him so that he cannot lick at it.
Thank you for asking Petco Pet Education Center, formerly Petcoach!