Is the method of bringing a glass of water and repeatedly uttering the name of an alleged envier, then dipping one's hand into the water after experiencing certain physical effects, in order to use this water to perform ghusl (ritual bath) from the effect of the envier, valid?

There is no reported evidence that merely mentioning the name of the envier (al-ain) is sufficient for treatment from the evil eye (ain). The fundamental principle is that the envier should wash, and the water from their washing should be poured over the afflicted person. If it is not possible for the envier to wash, then recourse should be made to ruqyah shariyyah (Islamic incantation), as the Prophet, peace and blessings be upon him, commanded seeking ruqyah for the evil eye. Jibreel (Gabriel), peace be upon him, came and performed ruqyah on him with: "In the Name of Allah, I perform ruqyah upon you, from everything that harms you, from the evil of every soul or envious eye, may Allah cure you. In the Name of Allah, I perform ruqyah upon you."