A non-Muslim woman needs only a Muslim wali when she has no other wali from her family. Under normal circumstances, the wali should be of the same religion as the bride. I could not find any distinction on this specific matter based on schools of jurisprudence, but I could be wrong. The differences that I found were related to pillars (requirement of a wali being for a mature, adult woman of sound mind), and conditions (multiple), and even in the order of who should be wali, but not the belief of the wali. I do not know if an English version exists for the book Al-Fiqh 'Ala Al-Madhahib Al-Arba'a (Arabic: الفقه على المذاهب الأربعة) by 'Abdul-Rahamn Al-Jaziri (Arabic: عبد الرحمن بن محمد عوض الجَزِيري) for access to more information on the topic.
The answer to question 39757 you are referring to partially quotes a fatwa by the Permanent Committee (Al-Lajnah Ad-Da'imah) Vol. 1, Chapter 18 (Book of Nikah), pp. 162, which is fatwa 11242. The conditions that need to be fulfilled in a wali are derived by scholars based on analogy:
- Male
- Adult
- Mature
- Of sound mind
- Free
- Of good character
- Shares common belief with the bride
An elaboration on the fatwa by Abd Al-Aziz Fawzan Al-Fawzan, a member of this Permanent Committee, may shed some light on this matter:
فلا ولاية لغير المسلم على المسلم، ولا لمسلم على غير المسلم ولا لمرتد على أحد مطلقا ،فإنه لا يرث غير المسلم من المسلم ولا المسلم من غيره، ولا المرتد من أحد؛ لأن اختلاف الدين من موانع الإرث، والولاية كالإرث كلاهما مستند إلى القرابة
Al-Fawzan states that both the wali and the bride have to be of the same religion. Non-Muslims have no wilaya (jurisdiction) over Muslims, and Muslims have no wilaya over non-Muslims. This is the analgous to inheritence [jursidiction over property of the deceased]: Non-Muslims do not inherit from Muslims, and Muslims do not inherit from non-Muslims. As difference of religions prohibits inheritance , then wilaya follows suit, as both are functions of kinship.
وفي خصوص المسلمة فلا يصح أن يتولى عقد نكاحها كافر ولو كان أباها
Al-Fawzan states that in the matter of a Muslim woman, it is not allowed to have her marriage contract by supervised by a disbeliever, even if this disbeliever is her father.
وأما الكافرة الكتابية يتزوجها مسلم ،فإنه يجوز لأبيها الكافر أن يعقد لها عليه لثبوت التوارث بينهما و لقوله تعالى: « والذين كفروا بعضهم أولياء بعض » الأنفال ٧٣. وقوله تعالى: « وَلَن يَجْعَلَ اللّهُ لِلْكَافِرِينَ عَلَى الْمُؤْمِنِينَ سَبِيلاً » (النساء ١٤١) فقد قطع الله الموالاة بين المؤمنين والكافرين ولو كان لها أخوان أحدهما كتابي والآخر مسلم وليس لها أب ولاجد، فالولاية للكتابي كذلك
Al-Fawzan explains that for a Muslim man getting married to a woman from the People of the Scripture (Jews or Christians), her father as a disbeliever is allowed to be her wali, as she can inherit from him, and because Allah says "And those who disbelieved are allies of one another." (Al-Anfal 73). Allah says: "and never will Allah give the disbelievers over the believers a way [to overcome them]." (An-Nisa 141), so Allah has severed the muwalah between believers and disbelievers. If this woman from the People of the Scripture has two brothers, one is a Muslim and the other is from the People of the Scripture, the one who is from the from the People of the Scripture is her wali.
ولا يكون المسلم وليا لكافرة فإن عقد لابنته الكافرة لكافر فلا نتعرض لفسخه وقد ظلم المسلم نفسه، وأما لو عقد لكتابية على مسلم فإنه يفسخ أبداً؛ أن المولى عليها غير مسلمة فيكون لقريبها غير المسلم ولاية تزويجها
Al-Fawzan further elaborates that a Muslim man cannot be the wali of a woman who is a disbeliever. If a Muslim man enters a marriage contract for his daughter who is a disbeliever to a man who is also a disbeliever, the marriage contract does not get invalidated, but the man has committed transgression. If a Muslim man enters a marriage contract for his daughter who is a disbeliever to a Muslim man, the marriage contract is null and void. Since the woman is a non-Muslim, only a non-Muslim can be her wali.
وقد استثنى العلماء من هذا الشرط الولاية العامة وذلك كولاية الحكم والقضاء لأنها تتناول المسلمين وغيرهم فيجوز للحاكم المسلم أو القاضي المسلم أن يزوج غير المسلمات إذا لم يكن لهن ولي نسيب لقوله ﷺ السلطان ولي من لا ولي له
Scholars exempted from this condition the general mandate of wilaya when replaced by an executive or a judiciary entity. It is permitted for a Muslim ruler or a Muslim judge to execute marriage contracts between a Muslim man and a non-Muslim woman based the hadith in (Sunan Ibn Majah, Book 9, Hadith 1953) that the ruler is the guardian of the one who does not have a guardian.
