Why Can't You Fast on Your Period?
Question

We would like to know the reason why menstruating women do not fast even though fasting has nothing to do with impurity.
Praise be to Allah.Muslims must submit to the rulings of Allah
The believer has to submit to the ruling of Allah and accept it even if he does not know the reason behind it; rather it should be sufficient for him that Allah and His Messenger have commanded it. Allah says (interpretation of the meaning): 
“It is not for a believer, man or woman, when Allah and His Messenger have decreed a matter that they should have any option in their decision. And whoever disobeys Allah and His Messenger, he has indeed strayed into a plain error.” [al-Ahzab 33:36] 
“The only saying of the faithful believers, when they are called to Allah (His Words, the Quran) and His Messenger , to judge between them, is that they say: ‘We hear and we obey.’ And such are the successful (who will live forever in Paradise).” [al-Nur 24:51]
There is wisdom behind every ruling
The believer should believe with certainty of faith that Allah is Wise and that He does prescribe anything except in accordance with His ultimate wisdom, and He does not enjoin anything except that which is in people’s best interests and He does not forbid anything except to protect people  from its harm and evil. How well Ibn Kathir put it in al-Bidayah wa'l-Nihayah (6/79) when he said:  
“The law of the Prophet (peace and blessings of Allah be upon him) is the most complete of laws; it did not leave any good thing that is recognized by the wise as being good but it enjoined it, and it did not leave any evil thing that is recognized by the wise as being evil but it forbade it. It has not enjoined anything of which people could say, ‘Would that this were not enjoined,’ and it has not forbidden anything of which people could say, ‘Would that this were not forbidden.’ 
We may understand the reasons, or they may be hidden from us, and most or part of them may be hidden from us. 
Can a menstruating woman fast?
The scholars unanimously agreed that it is forbidden for a menstruating woman to fast , and that she has to make up the days that she misses because of menstruation if that was an obligatory fast, as in Ramadan. 
They also agreed that if she does fast, her fast is not valid . 
Why can't a woman fast on her period?
The scholars differed as to the reason why the fast of a menstruating woman is not valid.  
Some of them said that the reason is not known to us. 
Imam al-Haramayn said: “We do not know why her fast is not valid, because purity is not a prerequisite for it.” (Al-Majmu’, 2/386) 
Others said: The reason why Allah has forbidden menstruating women to fast at the time of their period is out of mercy towards them, because loss of blood weakens them and if a woman were to fast when menstruating, she would be weakened both by menstruation and by fasting, in which case fasting would be an unfair burden and may even be harmful. 
Shaykh al-Islam [Ibn Taymiyah] said in Majmu’ al-Fatawa (25/234): 
“With regard to menstruation we say: 
Islam brings moderation in all things, and going to extremes in worship is a kind of unfairness that the Lawgiver forbids, and commands us to be moderate in worship. Hence Islam tells us to hasten breaking the fast and to delay suhur, and it forbids continual fasting. The Prophet (peace and blessings of Allah be upon him) said: “The best and most moderate of fasting is the fast of Dawud (peace be upon him). He used to fast alternate days and he would not flee when meeting the enemy.” Moderation in worship is one of the greatest aims of the Lawgiver. Hence Allah says (interpretation of the meaning): 
“O you who believe! Make not unlawful the Tayyibat (all that is good as regards foods, things, deeds, beliefs, persons) which Allah has made lawful to you, and transgress not. Verily, Allah does not like the transgressors.” [al-Maidah 5:67]
Forbidding permitted things is regarded as a transgression which goes against moderation. And Allah says (interpretation of the meaning):
“For the wrongdoing of the Jews, We made unlawful for them certain good foods which had been lawful for them — and for their hindering many from Allah’s Way; And their taking of Riba (usury) though they were forbidden from taking it.” [al-Nisa 4:160]
Because they went astray, they were punished by being forbidden good things, unlike the ummah that follows the middle way, for whom good things are allowed and bad things are forbidden. As this is the case, the fasting person is forbidden to eat and drink things that strengthen him and nourish him, and he is forbidden to deliberately cause his body to eject things that may weaken him. Otherwise if he were allowed to do that, he would be a transgressor and one who goes to extremes in his worship, and he would not be moderate. 
Things that come out of the body are of two types. The first type is those that are unavoidable or which come out in a manner that does not cause any harm. There is nothing wrong with these, like urine and stools, because they do not cause any harm and they cannot be avoided. If they need to come out, that does not harm a person, rather it benefits him; the same applies if a person vomits and cannot help it, or experiences a wet dream, which also cannot be helped. But if a person vomits deliberately and ejects the food and drink that give him nourishment, or masturbates accompanied by feelings of desire.
A menstruating woman can fast at a time other than the times when she is bleeding, because then she will be fasting at a time when she is in a good state, when she is not losing blood which gives strength to the body. Fasting at the time when she is bleeding would weaken her physically and would mean that she is fasting at a time when she is not in a good state, so she is commanded to fast at times other than menstruation.” 
And Allah knows best.