You may want to use eye drops for a while. It may lessen the redness of your eye while you wait until your appointment date.
There might be something causing strain to your right eye, which is why it could be red. There are times where we don't think anyone is wrong with our eye but there is and only the eye doctor can see it.
Take me for instance. I don't feel any pain in my eyes whatsoever but my eye doctor sees a thinning along the surface when they take pictures.