We cannot speak to the law in Georgia in particular, but we can discuss the situation and laws in general. You are facing a difficult situation with conflicting rights and interests. On the one hand, you have the right to control who is in your home and have an interest in protecting your stepmother. On the other, she has the right to see who she wants and her grandson should be free to visit his grandmother.
Your stepmother has the right to determine who she sees and who visits her in her home. If she says she wants her grandson to come visit, he should be allowed to do so. Since she has Alzheimer’s disease, she may not be able to make a determination one way or another and technically cannot say whether or not she wants her grandson to come visit. If you were appointed her guardian, you could make this decision for her. But if you are not her official guardian, then you must balance your interests in keeping this guy out of your house with his interest in visiting his grandmother. There is no clear legal answer without going through a guardianship proceeding. For information on guardianship, go here: http://www.elderlawanswers.com/guardianship-and-conservatorship-12096.
The best thing to do is to try to work out a compromise. Either he should be able to visit -- with some clear agreed-upon guidelines (which should be in writing so that there’s no disagreement about what they are) -- or you should agree to bring your stepmother to a neutral place where they can visit with one another. Another reason to work out a compromise is that caregivers who don’t facilitate visits by family members often get accused of all sorts of things – maltreatment, bad care, seeking the elder’s money – which may not be true, but which grow in the imagination of people who don’t see the real situation. If you have nothing to hide, then be totally transparent and permit visits. However unpleasant this grandson’s presence may be, it may avoid more unpleasantness in the future. That said, there are, of course, no guarantees either way.