What is the ruling of Islam on greeting by waving the hand when a person is far away, and is there any basis for this in Sharia?

Greeting by gesturing without uttering "peace" (Salam) resembles the Jews and Christians. This is indicated by the saying of the Messenger of Allah, peace and blessings be upon him: "He is not of us who imitates others. Do not imitate the Jews or the Christians, for the greeting of the Jews is by pointing with fingers, and the greeting of the Christians is by pointing with palms." And his saying: "Do not greet with the greeting of the Jews, for their greeting is by gesturing with heads and palms." The prohibition of greeting by gesturing is specific to those who are able to utter "peace" (Salam). As for those who are unable to utter "peace" (such as a person praying, someone far away, the mute, and the deaf), gesturing along with the utterance, if possible, is legislated for them. Scholars have explicitly stated that greeting by gesturing without the utterance of "peace" (Salam) is an innovation.