Move any furniture away from the wall and place it in another room. Place a drop cloth under the affected area.
Square the hole in the textured wall by using a saw to make the sides even and the same length and width. Measure the squared hole with a tape measure.
Put on eye protection and a dust mask.
Cut a piece of drywall or Sheetrock the same size of the square to fill the hole. Secure the patch to the wall studs by fastening the patch with drywall screws using a screw gun.
Tape the four joints with drywall tape and then rub joint compound over the entire patch with a putty knife, including the drywall tape. Allow to dry according to the manufacturer's recommendations.
Sand the patch and joints with sandpaper to even it with the rest of the textured wall. Apply a coat of spray-on plaster to the patch to match the rest of the textured wall. Allow the spray-on plaster to set as directed by the manufacturer.
Take the cut-out pieces from the old plaster to a hardware, paint or home improvement store and have the paint color matched. Paint over the textured plaster and let dry.