As with any topic within the discipline of history, there are no straight lines that can be drawn all the way back to a singular moment that explains how we got to where we are currently with the issue of reproductive rights and access to abortion. It has been popular as of late to explain the history of abortion quickly and neatly by stating that the pro-life movement began during the Civil Rights Era within churches that were looking to replace the issue of race and segregation that divided Christian dominations with the issue of abortion. I noticed this trend here with some of the deleted comments While that is true to some extent, it does not paint a total picture of the history.

Why is it important to look past Roe v. Wade and the beginnings of the New Right in the Goldwater/Nixon/Reagan era? Well, because abortion had been illegal in many states before that famous Supreme Court case (Roe v. Wade) that laid that national groundwork for how abortion has moved forward since. Before that, however, it was left up to the courts and legislature of the states.

Perhaps surprising many people today, large swaths of Protestant, Methodist, and Baptist clergy and leadership did not have strong public stances on abortion leading up to the mid 20th century and the politicalizing of the abortion ‘issue’ in the United States. The Catholic church, on the other hand, had explicitly denounced the practice in the late 1800s. There have been some ebbs and flows in religious thought about abortion being a sexual sin versus being viewed as a homicide/infanticide or declaring abortion a sin if it was used to cover up adultery (a trend many notice in feminist thought as being evidence of mans continually involvement in women's sexual and reproductive rights). “Early Church leaders began the debate about when a fetus acquired a rational soul, and St. Augustine declared that abortion is not homicide but was a sin if it was intended to conceal fornication or adultery.” [1]

Many denominations supported the idea of abortions to save the mother's life. Lack of significant public stances from churches could have been, of course, because the issue was not seen as explicitly necessary given its widespread illegality at the time, but abortion has been around for thousands of years. Thus, churches did have opinions on them, but not in the way that we are familiar with today (for example, denominations taking hardline public approaches and creating single-issue voting topics divided largely by political affiliation or ideology). This isn't a history of abortion opinions within the dominant religions in the United States, it is important to recognize that while the majority of Christians at least viewed abortion as wrong, the issue just wasn’t at the forefront of political and religious ideology.

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That said, religion and morality have always been tightly interwoven when it comes to women’s rights in the United States. Take, for example, the infamous Comstock Laws that were codified by congress. These ‘anti-vice’ laws banned explicit material, among other things, and banned the publication of information about contraceptives. Famously, the founder of what would become Planned Parenthood, Margaret Sanger, would be tried under these laws for providing access and publications of contraceptives. While Sanger was against the practice of abortion, she was in support of expanding women’s reproductive rights and this push for contraceptives helped pave the way for later generations of women’s rights activists.[2] This is a good time to note the importance of focusing beyond the New Right movement of Goldwater/Nixon/Reagan in terms of abortion. Abortion and the rights of women to control their bodies is an extremely old issue. It was illegal for Sanger in 1916 to even open a contraceptive clinic in the United States. So, the issue is exceptionally deep. However, for the sake of time/and I know I’m rambling, the issue did gain new traction during the rise of modern-day Conservatism in the United States.

Again, surprising to some, “Reagan had signed the Therapeutic Abortion Act in 1967, a reform bill proposed after a group of physicians were indicted for performing abortions on women who had rubella.”[3] So where did the flip happen with someone like Reagan (a conservative icon) supporting limited abortions to abortions being a politically divisive issue? Historian Jennifer Holland, for example, argues that pro-life advocacy can also be tied in with the push to preserve the nuclear family (family values), morality, Christianity, and the belief in the ideal American life. The ideal “American” life being contrasted against the Soviet Union and Communism during the Cold War.

4LeafTayback's response does a good job of a pointing toward the multi-faceted explanation that any decent response to this question ought to include. It is hard (if not impossible) to pinpoint some exact moment or event in which contemporary American Evangelical views on abortion (or most any social issue for that matter) suddenly took shape; they didn't appear suddenly, but gradually over time due to a number of developing social and political circumstances. Many of the dominant Evangelical views today on a variety of social issues developed alongside, and in interaction with, the broader form of American conservatism that emerged in the latter half of the 20th century (a movement often considered to be typified by the presidency and political landscape of the Reagan years in the 1980's). One great source on this topic is Ronald Story and Bruce Laurie's The Rise of Conservatism in America, 1945-2000. Their book contains a number of primary source documents (speeches, newspaper articles, academic papers, etc.) related to the conservative movement in the United States, and among these are several documents related to the rise of the "religious right" and the "moral majority"; many contemporary views (Evangelical and otherwise) related to anti-abortion positions developed within the context of this larger movement.

This contextualization should help drive home the point that Evangelical views on abortion didn't arise independently of the views of other religious groups. The anti-abortion views of the "moral-majority alliance" of the 70's and 80's; an alliance of which Evangelicals were certainly a part of; included, in the words of one of its key leaders Paul Weyrich (a Catholic) "ethnic Catholics, Gospel-believing protestants, Mormons, and Orthodox Jews working together" (Paul Weyrich, "Building the Moral Majority," Conservative Digest, August 1979, 18-19). For example, Catholic views against abortion in particular, though certainly appearing more prominently in the public sphere after Roe vs. Wade, were present long before the 1973 Supreme Court decision. The intermixing of the views of these various religious groups, contextualized by and in contact with the larger conservative movement in the United States, all contributed the development of the prominent views held by many Evangelicals today relating to abortion.

It should be pointed out also that when trying to accurately account for contemporary anti-abortion views held by Evangelicals (and others more generally), one should make a distinction between causation and correlation as regards the reasoning behind these beliefs. Though often associated with religion, anti-abortion views are held by a not-insignificant number of individuals whose opposition is not explicitly grounded in a particular religious belief. Groups like Secular Pro-Life argue that abortion isn't primarily a religious issue, but rather a philosophical one related to natural rights, and therefore able to be discussed/debated without any appeal to religious doctrine. Even religious people who hold anti-abortion views don't necessarily hold them because of their religious beliefs. Robert Spitzer, S.J., a Jesuit priest and philosopher, argues this position in his book Ten Universal Principles: A Brief Philosophy of the Life Issues; though he does hold religious views, he argues that opposition to abortion need not (in fact, ought not) be primarily grounded in religious doctrine, but rather in reasoned philosophical and scientific debate.

I point this out in order to try and account for the entire spectrum of (and reasoning behind) anti-abortion views visible today. Your question seems to imply that contemporary anti-abortion views are held only as a result of some sort religious belief. No doubt, a significant number (and almost certainly, I suspect, the majority) of Evangelicals and others who are opposed to abortion hold their views primarily because of their religious convictions. That said, I would argue that it is erroneous to think of all people that are opposed to abortion as necessarily being especially religious, and more specifically, that it is erroneous to think that if someone is anti-abortion and religious, that they necessarily hold their anti-abortion views primarily because of their religious beliefs. This might help account (to some extent, at least) for the existence of those who might hold anti-abortion views in spite of the fact that, as you mention, there is no (as far as I am aware) explicit mention of abortion in biblical literature.