1. Biblical Origins of a Distinctly Jewish View
וכי ינצו אנשים. זֶה עִם זֶה, וְנִתְכַּוֵּן לְהַכּוֹת אֶת חֲבֵרוֹ וְהִכָּה אֶת הָאִשָּׁה: עברית
Rashi on 21:22
IF MEN STRIVE with each other and one intended to strike the other and inadvertently struck the woman (Sanhedrin 79b).
The Christian tradition reads this differently. In the Septuagint, the Greek translation of the Bible, the Hebrew word for "no harm follow" was replaced by the Greek for "her child to be born perfectly formed." This translation/interpretation brands the unborn child affected by this violence as fully formed and therefore sees the action against it as murder/manslaughter. The Jewish tradition followed Rashi as seeing the harm described about the woman and not the unborn child.
2. Is abortion a capital offense? A moral offense? Conflicting Views
(6) Whoever sheds the blood of man, By man shall his blood be shed; For in God's image Did God make man.
משום רבי ישמעאל אמרו אף על העוברין מאי טעמיה דרבי ישמעאל דכתיב (בראשית ט, ו) שופך דם האדם באדם דמו ישפך איזהו אדם שהוא באדם הוי אומר זה עובר שבמעי אמו
עברית
In the name of Rabbi Yishmael it was said: [A Noachide may be put to death] even for the killing of a fetus. What is Rabbi Yishmael’s source? For it is written, “One who spills the blood of a person shall have his own blood spilled by another person [literally read: one who spills the blood of a person inside another person etc.].” (Bereishit 9:6). Which person is inside another person? This is referring to the fetus in the mother’s womb.
(ד) הָאִשָּׁה שֶׁהִיא יוֹצְאָה לֵהָרֵג, אֵין מַמְתִּינִין לָהּ עַד שֶׁתֵּלֵד. יָשְׁבָה עַל הַמַּשְׁבֵּר, מַמְתִּינִין לָהּ עַד שֶׁתֵּלֵד. הָאִשָּׁה שֶׁנֶּהֶרְגָה, נֶהֱנִין בִּשְׂעָרָהּ. בְּהֵמָה שֶׁנֶּהֶרְגָּה, אֲסוּרָה בַהֲנָיָה:
(4) If a woman is about to be executed, they do not wait for her until she gives birth. But if she had already sat on the birthstool, they wait for her until she gives birth. If a woman is executed, one may use her hair. If an animal has been executed, it is forbidden to make any use of it.
Another source in this area is Exodus 21:12 which states, "A person who fatally strikes a man shall be put to death." Rashi explains that this does not apply to unborn children.
3. Jewish View on Stages of Pregnancy and When Life Begins
יבמות ס״ט ב:ל״ו
טובלת ואוכלת עד ארבעים דאי לא מיעברא הא לא מיעברא ואי מיעברא עד ארבעים מיא בעלמא היא
עברית
If she is found pregnant, until the fortieth day it is only a mere fluid.
(ז) הַמַּפֶּלֶת לְיוֹם אַרְבָּעִים, אֵינָהּ חוֹשֶׁשֶׁת לְוָלָד. לְיוֹם אַרְבָּעִים וְאֶחָד, תֵּשֵׁב לְזָכָר וְלִנְקֵבָה וּלְנִדָּה. רַבִּי יִשְׁמָעֵאל אוֹמֵר, יוֹם אַרְבָּעִים וְאֶחָד, תֵּשֵׁב לְזָכָר וּלְנִדָּה. יוֹם שְׁמוֹנִים וְאֶחָד, תֵּשֵׁב לְזָכָר וְלִנְקֵבָה וּלְנִדָּה, שֶׁהַזָּכָר נִגְמָר לְאַרְבָּעִים וְאֶחָד, וְהַנְּקֵבָה לִשְׁמוֹנִים וְאֶחָד. וַחֲכָמִים אוֹמְרִים, אֶחָד בְּרִיַּת הַזָּכָר וְאֶחָד בְּרִיַּת הַנְּקֵבָה, זֶה וָזֶה לְאַרְבָּעִים וְאֶחָד:
(7) If she miscarries on the fortieth day [since her prior immersion], she need not be concerned that it was a fetus. If [she miscarries] on the forty-first day, she should sit [for the required number of days] for a male and for a female, and for [being] a niddah. Rabbi Yishmael says: on the forty-first day she should sit for [the required number of days for one who gives birth to] a male and for [being] a niddah. On the eighty-first day she should sit for a male and for a female and for [being] a niddah, because [the formation of] a male is completed in forty-one [days], and a female in eighty-one. And the Sages say: this and that [are both completed] in forty-one.
Rabbi Yehuda HaNasi holds: A fetus is considered as its mother’s thigh, i.e., a part of its mother’s body, and it is as though the master transferred ownership of one of her limbs to her.
(ג) תנוקת בת יום אחד מיטמאה בנדה בת עשרה ימים מיטמאה בזיבה. תנוק בן יום אחד מיטמא בזיבה. ומיטמא בנגעים. ומיטמא בטמא מת. וזוקק ליבום ופוטר מן היבום ומאכיל בתרומה ופוסל מן התרומה ונוחל ומנחיל. וההורגו חייב. והרי הוא לאביו ולאמו ולכל קרוביו. כחתן שלם:
עברית
(3) A baby girl one day old can become impure as a niddah. At ten days old she can become impure as a zavah A baby boy one day old.... can become impure as a zav, or impure by having nega'im, or impure through [contact with] the impurity of a corpse; and he creates a bond for yibum [the obligation to perform Levirate marriage]; and he can exempt one from requiring yibum and he can feed terumah and he can render terumah to be forbidden and he can inherit or cause others to inherit and one who kills him is liable [for murder]; and he is thereby fully a family member to his father and to his mother and for all his relatives.
English
It seems to me that the widespread opinion is in favor of full life beginning at birth, with some stages of development along the way. I also found a few sources that challenged this to some degree. One was a story of a Greek officer who convinces Rabbi Yehudah HaNasi to change his mind about when life begins! Another was a teaching by Nachmanides (medieval commentator) that we should violate Shabbat for the sake of an unborn baby.
4.When (and WHY) Abortion is Permitted
(ו) הָאִשָּׁה שֶׁהִיא מַקְשָׁה לֵילֵד, מְחַתְּכִין אֶת הַוָּלָד בְּמֵעֶיהָ וּמוֹצִיאִין אוֹתוֹ אֵבָרִים אֵבָרִים, מִפְּנֵי שֶׁחַיֶּיהָ קוֹדְמִין לְחַיָּיו. יָצָא רֻבּוֹ, אֵין נוֹגְעִין בּוֹ, שֶׁאֵין דּוֹחִין נֶפֶשׁ מִפְּנֵי נָפֶשׁ:
(6) A woman who was having trouble giving birth, they cut up the fetus inside her and take it out limb by limb, because her life comes before its life. If most of it had come out already they do not touch it because we do not push off one life for another.
..יצא ראשו אין נוגעין בו לפי שאין דוחין נפש מפני נפש ואמאי רודף הוא שאני התם דמשמיא קא רדפי לה נימא מסייעא ליה רודף שהיה רודף אחר חבירו להורגו אומר לו ראה שישראל הוא ובן ברית הוא והתורה אמרה (בראשית ט, ו) שופך דם האדם באדם דמו ישפך אמרה תורה הצל דמו של זה בדמו של זה
עברית
'If his head emerged' we do not touch him since we do not push away one life for another. Why? It is a pursuer (rodef)....A pursuer who was chasing another to kill him, say 'He is a Jew and a child of the covenant.' But the Torah said, 'The one who sheds blood, shall his own blood be shed.', meaning, save the blood of the pursued by the blood of the pursuer.
רש"י על סנהדרין ע״ב ב:ל״ח:א׳
יצא ראשו - באשה המקשה לילד ומסוכנת וקתני רישא החיה פושטת ידה וחותכתו ומוציאתו לאברים דכל זמן שלא יצא לאויר העולם לאו נפש הוא וניתן להורגו ולהציל את אמו אבל יצא ראשו אין נוגעים בו להורגו דהוה ליה כילוד ואין דוחין נפש מפני נפש
עברית
its head came out: With a women that is experiencing difficulty giving birth and is in danger. And it is taught in the first section [of this teaching], "the midwife extends her hand and cuts it up and extracts [the pieces];" as the entire time that that it has not gone out into the environment of the world, it is not a soul, and [so] it is possible to kill it and to save its mother. But when its head came out, we cannot touch it to kill it, as it is like a born [baby]; and we do not push off one soul for the sake of another.
English
(ט) אַף זוֹ מִצְוַת לֹא תַּעֲשֶׂה שֶׁלֹּא לָחוּס עַל נֶפֶשׁ הָרוֹדֵף. לְפִיכָךְ הוֹרוּ חֲכָמִים שֶׁהָעֻבָּרָה שֶׁהִיא מַקְשָׁה לֵילֵד מֻתָּר לַחְתֹּךְ הָעֵבָּר בְּמֵעֶיהָ בֵּין בְּסַם בֵּין בְּיָד מִפְּנֵי שֶׁהוּא כְּרוֹדֵף אַחֲרֶיהָ לְהָרְגָהּ. וְאִם מִשֶּׁהוֹצִיא רֹאשׁוֹ אֵין נוֹגְעִין בּוֹ שֶׁאֵין דּוֹחִין נֶפֶשׁ מִפְּנֵי נֶפֶשׁ וְזֶהוּ טִבְעוֹ שֶׁל עוֹלָם:
(9) ... It is a negative commandment that one should not protect the life of a rodef (pursuer). For this reason, the sages ruled that in the case of a pregnant woman in a dangerous labor, it is permissible to dismember the fetus in her womb - whether with a drug or by hand because it is like a rodef pursuing her to kill her. However, once his head has emerged one may not touch him, as we do not set aside one nefesh [soul] for another, and this is the natural way of the world.
Excerpts from Igrot Moshe (Reb Moshe Feinstein, 1895 – 1986)
It would be forbidden to kill it even to save someone’s life. The exception would be to save the life of the mother during childbirth, not for any other need of the mother, which would definitely be forbidden.
Even for children for whom the doctors predict a very short life span, such as those children who are born with the disease called Tay-Sachs, which through newly developed tests can be diagnosed prenatally, it would be forbidden since there is no danger to the mother and the infant is not a rodef. One cannot permit an abortion even though there is very great suffering involved … It is incontrovertible and clear as I have written, a straightforward halachah according to the words of our Masters, the traditional commentaries and halachic authorities, that abortion would be forbidden as bona-fide murder, for any fetus; legitimate or a mamzer, genetically normal or afflicted with Tay-Sachs, are all included in the prohibition according to Jewish law.
Excerpts from Tzitz Eliezer (Eliezer Waldenberg, Jerusalem, 1915 – 2006)
If there is a danger to the mother from continuing the pregnancy, one should permit abortion without hesitation. Also, if her health is poor and to cure her or to relieve her from great pain it is necessary to abort the fetus, even if she is not in actual danger, there is room to permit it, based on the halachic authority’s evaluation of the situation.
In the case of a baby who will have Tay-Sachs, "One should permit...abortion as soon as it becomes evident without doubt from the test that, indeed such a baby shall be born...if, indeed, we may permit an abortion according to the halachah because of 'a great need' and because of pain and suffering, it seems that this is the classic case for such permission. And it is irrelevant in what way the pain and suffering is expressed, whether it is physical or psychological. Indeed, psychological suffering is in many ays much greater than the suffering of the flesh.
Responsum L'vushai Mord'khai, 1913
Mental health risk has been definitely equated to physical-health risk. This woman who is in danger of losing her mental health unless the pregnancy is interrupted, therefore, would accordingly qualify.
5. Tradition has a vote but not a veto: Other Considerations from Jewish Values and Tradition
B'tselem Elohim- the dignity not just of the fetus and potential life but of the woman
Autonomy - a more modern concept but the right to self-determination
Other ideas?