The woman can ask but the torah specifies that the man must write (order the writing of) the divorce document and deliver it to her. There are cases in which the court can order the man to write the get, but he must be the one to write it (or order it written) and it must be of his own free will. This is similar to the rules of getting married in which the man must give the woman the ring (or other object) in order for the wedding to be valid.
Rav Hirsch points out that the methodology of a divorce must be similar to the methodology of the marriage. Just as the man must give the wife an object and "take" her into his house, similarly he must be the one to sever that relationship.
The first act קידושין ... is the personal adoption of the wife by the
husband. ... and is further explained in Kidushin 2a, this adoption is
made in the form of an act of appropriation
... the complete unity of the wife with the person of her husband is
not yet completed, but is conditional on another act ... chupah
This means that just as the Bnei Yisrael at Har Sinai gave themselves up to the authority of the Torah, the woman has given up her authority to separate from the husband and the marriage can only be ended by death and divorce.
All of these laws can be summarised in one idea: the written word is
to be used in the place of the spoken one, instead of a speech made
with full considered conscious intention, the thought of divorce must
be expressed in writing with equal import of its import and
... By depositing the deed of divorce in any place that is her domain
... the husband gives up any rights he has in it ...
Further Rav Hirsch points out
ושלחה מביתו is simply the expression used here for the legal effect of
the act of divorce: the freeing of the marriage bond which hitherto
was in force, and the releasing of the belonging to the husband
This is exactly the inverse of the sealing of the original marriage bond.
Rav Hirsch also points out it was after more than two thousand years that there was on the whole no abuse of that authority and it was not "progress" that it was found necessary to curtail that authority to protect the woman. He then refers to an article that he wrote on the Position of Jewish Women. See the article in volume 2 of "Judaism Eternal".
This means that the woman cannot "decide on her own" to divorce. Even in cases in which the woman can go to bais din to force the husband to grand a divorce, the husband must still write and deliver the get. The technical methodology requires that the initiative come from the husband.
As an example, we see the Rambam. Note that this does not say that the halacha of maus is accepted by everyone. I am giving it to show that even in this case, the technical requirement still applies.
The Rambam Hilchos Ishus 14:8 points out that a woman can force the divorce to occur by acting in such a way as to be in the status of "moredes", that is, a rebellious wife, such as taking a vow to have no relations with the husband. In that case, the husband must divorce her, though she does not receive the kesuvah. However, the technical methodology requires that the initiative come from the husband.
A woman who withholds marital intimacy from her husband is called a
moredet ("a rebel"). She is asked why she has rebelled. If she
answers: "Because I am repulsed by him and I cannot voluntarily engage
in relations with him," her husband should be compelled to divorce her
immediately. For she is not like a captive, [to be forced] to engage
in relations with one she loathes.14
[In such an instance, as part of] the divorce [settlement], she does
not receive any of the money promised her in her ketubah.15
Note that Rambam Hilchos Gerushin 2:20states that the court can order the man coerced until he says "I am doing this of my own free will" because of the principle that any Jew wants to obey the Torah. However, if he does not say this, the divorce is invalid.
When a man whom the law requires to be compelled to divorce his wife51
does not desire to divorce her, the court should have him beaten until
he consents, at which time they should have a get written. The get is
acceptable. This applies at all times and in all places.52
Similarly, if gentiles beat him while telling him: "Do what the Jews
are telling you to do," and the Jews have the gentiles apply pressure
on him until [he consents] to divorce his wife, the divorce is
acceptable. If, however, the gentiles compel him to write [a get] on
their own initiative, the get is [merely] unacceptable.53The rationale
is that the law requires him to give a divorce.
Why is this get not void? For he is being compelled - either by Jews
or by gentiles - [to divorce] against his will [and a get must be
Because the concept of being compelled against one's will applies only
when speaking about a person who is being compelled and forced to do
something that the Torah does not obligate him to do - e.g., a person
who was beaten until he consented to a sale,54 or to give a present.
If, however, a person's evil inclination presses him to negate [the
observance of] a mitzvah or to commit a transgression, and he was
beaten until he performed the action he was obligated to perform, or
he dissociated himself from the forbidden action, he is not considered
to have been forced against his will. On the contrary, it is he
himself who is forcing [his own conduct to become debased].55
With regard to this person who [outwardly] refuses to divorce [his
wife] - he wants to be part of the Jewish people, and he wants to
perform all the mitzvot and eschew all the transgressions; it is only
his evil inclination that presses him. Therefore, when he is beaten
until his [evil] inclination has been weakened, and he consents [to
the divorce], he is considered to have performed the divorce