Can you suture if the wound is 24 hours old?

I cut the area between my thumb and index finger open the other night with a new knife. The wound is pretty deep and I went to the ER right away. After waiting for 4 hours I decided I would just make an appointment with my family doctor for the next day. When I saw the doctor the following day, she said that stitching the wound up 24 hours after the accident would not help the wound heal. I guess it's already started healing? So they cleaned the wound and told me to keep in covered. Now I'm worried about infection. It's ridiculous, you have to ride an ambulance to the hospital to get any attention.