All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah and that Muhammad is His slave and Messenger.
It is confirmed in the Sunnah that it is permissible to only use istijmaar (cleaning oneself with stones or another material in its place, such as tissue) without using water after defecating or urinating until the impurity itself is removed. Istijmaar is with stones and with something alike, like every pure solid that removes the very impurity and that is not something respected, such as handkerchiefs and pieces of cloth; Shaykh Al-'Uthaymeen said, “It is sufficient to do istijmaar with a handkerchief, and this is acceptable because the purpose of istijmaar is to remove the impurity; this is whether it is with handkerchiefs, pieces of cloth, earth (soil) or stones; however, it is not permissible for a person to do istijmaar with what is forbidden in the sharee'ah, such as bones and dung.”
As regards the issue of whether or not it is ruled that the place of impurity has become pure by istijmaar, then the scholars have two opinions in this regard; some of them said that the place of impurity does not become pure with istijmaar but that this is excused by the sharee'ah; and some of them said that the place of impurity becomes pure; Ibn Qudaamah said regarding the impurities that are excused:
“One of which is the place of Iitinjaa (cleaning oneself with water); this is excused from the effect of istijmaar after the impurity is removed, and fulfilling the required number [in using the item for istijmaar], without any difference of opinion that we know of, but the scholars of our School differed in opinion about its purity; Abu Abdullaah ibn Haamid and Abu Hafs ibn Al-Maslamah hold the view that it is pure. This is the apparent view of Imaam Ahmad as he said about a person who does istijmaar and who has sweat in his trousers: it is acceptable. Had this been impure, Imaam Ahmad would have considered it (sweat) impure. The evidence for this is that the Prophet said about dung and bones of dead animals that they do not become pure; this necessarily implies that other items than the dung and bones of dead animals become pure. That is also because it (istijmaar) is a means of removing the ruling of impurity, so it removes it like water.
Nonetheless, the late scholars of our School said: The place of impurity does not become pure; rather it is impure. This is because if a person who had performed istijmaar sits in little amount water, he would render it impure, and if he sweats, his sweat is impure, because wiping does not completely remove the parts of impurity, so what remains of it is impure, because it is the impurity itself, so it is like if this impurity exists in the place of impurity only.”
The Fiqh Encyclopedia reads:
"The scholars differed in this regard into two opinions: The first: that the place of impurity becomes pure with Istijmaar; this is one view of the Hanafi, Maaliki and Hanbali Schools of Fiqh...The second, which is the second view of the Hanafi and Maaliki Schools and the view of the late scholars of the Hanbali School: the place of impurity is impure but it is excused due to hardship..."
Even according to the view that it is not ruled to be pure, the hand does not become impure when it comes in contact with the place of impurity as long as the hand is dry because what remains in the place of impurity has the ruling of impurity and not the impurity itself, but if the hand is damp, it becomes impure.
For more benefit, please refer to fataawa 84603 and 173131.
Allaah knows best.