What is the ruling regarding giving the Salam with your hand [only]?

It is not permissible to give Salam by indicating with your hand [only]. The Sunnah is to speak it when giving it and replying to it. Giving the Salam by indicating with your hands is not permissible because it is an imitation of some disbelievers, and is not in accordance to what Allah legislated. However, there is no problem if one indicates with their hand, while [also] speaking it, to a person who is far away to show that they are giving Salam, because narrations support this. Likewise, if the person being greeted is praying, they respond by indicating with the hand [only], as was authentically reported in the Sunnah about the Prophet - ﷺ.