Yes, as long as it is legally defined as Istinjaa.
عَنْ عَائِشَةَ، أَنَّ رَسُولَ اللَّهِ صَلَّى اللهُ عَلَيْهِ وَسَلَّمَ قَالَ: «إِذَا ذَهَبَ أَحَدُكُمْ إِلَى الْغَائِطِ، فَلْيَذْهَبْ مَعَهُ بِثَلَاثَةِ أَحْجَارٍ يَسْتَطِيبُ بِهِنَّ، فَإِنَّهَا تُجْزِئُ عَنْهُ»
(Sayidah) Aishah narrated that the Messenger of Allah (ﷺ) said, ‘When one of you goes to relieve himself, he should take with him three stones with which to cleanse himself, for they will be sufficient for him.’ (Abu Dawood 40, Hasan Lighayrihi).
We see from the Hadith above that Istinjaa is valid using stones alone (without using water). Toilet paper or other such materials can take the place of stones and therefore carry the same ruling. However, this ruling is only applicable if the filth has not spread and cleaning it can therefore be classified as Istinjaa.
والاستنجاء سنة يجزي فيها الحجر وما يقوم مقامه يمسحه حتى ينقه وليس فيه عدد مسنون وغسله بالماء أفضل فإن تجاوزت النجاسة مخرجها لم يجز فيه إلا الماء (مختصر القدوري)
Istinjaa is Sunnah, using stone and the like suffices, one cleanses until it is clean. There is no particular Sunnah with regards to the number. Washing with water is better. However, if the impurity spreads beyond the orifice, then nothing but water is permitted (in order to remove the impurity). (Imam Abu al-Husain al-Quduri, Mukhtasar al-Quduri).
The Nass (reference) above differentiates between filth that has not spread beyond the orifice (which requires Istinjaa) and that which has spread. If it has spread, then it no longer takes the ruling of Istinjaa and thus requires water.
So, in conclusion we can say that, if the filth has not spread a lot then it is defined as Istinjaa and thus cleaning it with toilet paper is valid. However, if it has spread then it is no longer classified as Istinjaa and requires water (or the like) to clean it.
This is actually a fairly complicated Masalah and students are obliged to study it as part of a comprehensive study of Fiqh, often Q and A is not helpful when it relates to complicated issues if one has not studied them more holistically.
And Allah (Most High) Knows Best.
– Answered by Shaykh Noorud-deen (10.06.2021)
Are you in a state of purity after Istinjaa without water?
Is it sufficient to learn the Deen through Q&A only?
See also (video):