Firstly:
Allah, may He be Exalted, says: {Regarding those of your women who have passed the age of menstruation, if you are uncertain [of the ruling], their waiting period is three months; that also applies to those who have not yet begun to menstruate} [At-Talaq 65:4].
Thus regarding the woman who has not yet menstruated because she is young, the Lawgiver has instructed her, if she is divorced, to observe `iddah for three months.
Ibn Kathir (may Allah have mercy on him) said:
Allah, may He be Exalted, says, explaining the `iddah of the postmenopausal woman – who is the one whose menses have ceased due to her age – that it is three months, instead of the three menstrual cycles that are required of one who still menstruates, as is indicated by the verse in Surat al-Baqarah.
The same applies to young women who have not yet reached the age of menstruation; their `iddah is the same as that of the postmenopausal woman, namely three months. Hence Allah says: {that also applies to those who have not yet begun to menstruate}.
End quote from Tafsir Ibn Kathir, 8/149.
By saying that marriage cannot be consummated with a young woman who has not yet menstruated, the questioner is conflating what is mentioned in this verse, {those who have not yet begun to menstruate} with what is mentioned in the verse in Surat al-Ahzab, {you have no reason to ask them to observe any waiting period [`iddah] for you}. That is not correct, because if a woman has not started to menstruate yet because she is still young, that does not necessarily mean that it is not possible to have intercourse with her; rather it may be possible to have intercourse with her, with no problem, if she is physically able to have intercourse and there is nothing to prevent it, even if she has not yet started to menstruate. In that case, her husband is not to be prevented from having intercourse with her.
This has been discussed previously in an earlier answer in which we quoted scholarly views regarding that, and we explained that the fact that the marriage contract is done with a minor does not necessarily mean that her husband must consummate the marriage with her straight away. Rather the marriage contract may be done, then consummation may or may not happen.
There was a discussion about the wisdom behind arranging the marriage of a minor girl, and that doing so is not permissible unless it serves her interests.
Secondly:
If a minor girl is divorced, then there must be one of two scenarios:
The first scenario is where her husband had intercourse with her, and in this case the wisdom behind the instruction for her to observe `iddah is obvious, which is to ascertain that there is no pregnancy.
That is because the absence of menses does not necessarily indicate that there is no pregnancy, because it is possible for a woman to become pregnant before her first menses occurs, as menses comes after ovulation and the uterus becoming ready for pregnancy. This is something that is confirmed by medical specialists.
Therefore the Lawgiver instructed the woman who has not yet menstruated to observe `iddah [in the event of divorce] as a precaution to protect lineages and prevent the mixing of lineages, and as a means of definitively verifying that there is no pregnancy.
The second scenario:
If the husband spent time alone with his wife but did not consummate the marriage with her, whether she is young or old, does she have to observe `iddah, as is the case for one with whom the marriage has been consummated, or does she not have to observe `iddah, as is the case for one with whom the marriage was not consummated?
The scholars differed concerning that.
Ibn Qudamah (may Allah have mercy on him) said:
If he was alone with her but did not have intercourse with her, then he divorced her, then according to the view of Ahmad, she must observe `iddah. That was narrated from the Rightly-Guided Caliphs, and from Zayd and Ibn `Umar; it was also the view of `Urwah, `Ali ibn al-Husayn, `Ata’, az-Zuhri, ath-Thawri, al-Awza`i, Is-haq, as-hab ar-ra’y, and ash-Shafa`i in his earlier view.
In his later view, ash-Shafa`i said: She does not have to observe `iddah, because the verse {O you who believe, if you marry believing women and then divorce them before the marriage is consummated, you have no reason to ask them to observe any waiting period [‘iddah] for you} [al-Ahzab 33:49] is a clear religious text, and because she has been divorced before the marriage was consummated, so this is like the case of one whose husband did not spend time alone with her.
However, our view that `iddah is obligatory is supported by the consensus of the Sahabah. Imam Ahmad and al-Athram narrated with their isnad from Zurarah ibn Awfa that he said: The Rightly Guided Caliphs ruled that once the curtain is drawn or the door is closed, the mahr becomes due and `iddah must be observed (in the event of divorce). Al-Athram also narrated it from al-Ahnaf, from `Umar and `Ali; and from Sa`id ibn al-Musayyab, from `Umar and Zayd ibn Thabit. These verdicts are well known and established, and no one objected to them, thus it became a matter of consensus.
End quote from Al-Mughni, 11/197-198.
What further supports the view that `iddah is obligatory in this case, is that even though in principle the main reason for the prescription of `iddah is to ascertain that there is no pregnancy, there are other reasons why it was prescribed. The clearest evidence for that is the fact that establishing whether a woman is pregnant or not may be established by her missing one menstrual period. So, for example, if a man buys a slave woman, it may be ascertained that she is not pregnant if she misses one menstrual period only, and there is no need to wait for three menstrual cycles as in the case of the `iddah of a woman who has been divorced.
It was narrated from Abu Sa`id al-Khudri that the Prophet (blessings and peace of Allah be upon him) said regarding the female captives of Awtas: “Do not have intercourse with a pregnant woman until she has given birth, or with one who is not pregnant until she has had one menstrual period.” Narrated by Abu Dawud, 2157; classed as authentic by al-Albani in Irwa’ al-Ghalil, 1/200.
The wisdom behind the `iddah for a woman who menstruates and a woman who does not menstruate is to give time for reconciliation and to avoid breaking up families that have been formed. Often the cause of the divorce is a fleeting dispute; if the spouses show patience for some time, their anger will subside and they will become reconciled. It is for this reason that the `iddah was made lengthy, and one menstrual cycle is not sufficient for that. Moreover, the wife is instructed to remain in the marital home and not to move out during the `iddah, such as going to her father’s house, because this closeness between the spouses plays a role in facilitating reconciliation.
Allah, may He be Exalted, says:
{O Prophet [and believers], when any of you want to divorce women, then divorce them at a time when their prescribed waiting period can properly start, and keep an accurate count of the waiting period; and fear Allah your Lord. Do not turn them [divorced women] out of their houses [during the waiting period], nor should they themselves leave, unless they are guilty of manifestly immoral conduct. Such are the limits ordained by Allah; whoever transgresses the limits ordained by Allah has wronged himself. You do not know; perhaps Allah will bring about some new development [leading to reconciliation]} [At-Talaq 65:1].
Al-Qurtubi (may Allah have mercy on him) said:
{Do not turn them [divorced women] out of their houses [during the waiting period]} means: the husband has no right to turn the wife out of the marital home so long as the `iddah has not yet ended, and it is not permissible for her to leave either, because of the husband’s rights, except in the case of real necessity.
End quote from Tafsir al-Qurtubi, 21/35.
Shaykh `Abd ar-Rahman al-Mu`allimi (may Allah have mercy on him) said:
If a man divorces his wife, he must not stop spending on her maintenance and accommodation. The wisdom behind that – and Allah knows best – is to leave the door open to reconciliation. In most cases, the accommodation that he is instructed to provide for her is his own house or nearby, and that is more likely to lead to reconciliation. During the `iddah, he may call to mind some memories whilst he is lying in his bed, so all that remains between him and her is to lift the curtain or knock on the door, whereas if he waits until morning, perhaps his interest will subside, so he will not take her back.
End quote from Athar ash-Shaykh `Abd ar-Rahman al-Mu`allimi, 17/627-628.
What confirms this wisdom is the fact that if the wife asks for separation from her husband by means of khul` and the husband agrees to that, this woman is required to observe an `iddah of only one menstrual period.
It was narrated from ar-Rubayyi` bint Mu`awwidh ibn `Afra’ that she was separated from her husband by means of khul` at the time of the Prophet (blessings and peace of Allah be upon him), and the Prophet (blessings and peace of Allah be upon him) instructed her, or she was instructed, to observe an `iddah of one menstrual cycle. Narrated by at-Tirmidhi, 115; he said: The hadith of ar-Rubayyi` is authentic, which says that she was instructed to observe an `iddah of one menstrual cycle.
The wisdom behind the `iddah of one who separates from her husband by means of khul` being one menstrual cycle is that in the case of khul`, the man cannot take the woman back during the `iddah; rather as soon as the khul` is concluded, his wife becomes a stranger to him, and it is not permissible for him to take her back except with a new marriage contract.
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said:
This khul` means that the woman is irrevocably separated from the husband, so it is not permissible for him to marry her again after that except with her consent. It is not like regular divorce (talaq), because talaq may be revocable and the husband may take his wife back during the `iddah without her consent.
End quote from Majmu` al-Fatawa, 33/152.
For more information about khul`, please see the answer to question no. 5163.
And Allah knows best.