I admit when I first saw the paper from Third Way arguing that repealing the Mexico City gag rule would reduce the number of abortions, I figured it was a clever bit of spin to assuage pro-life Obama supporters, irritated that he’d overturned the ban.
But I have to admit, their memo on this is quite thought provoking. So far the pro-life folks are viewing Obama’s step as harshly pro-abortion, with Bill Donohue adding that it’s racist too. (“Here we have a black president taking money from the taxpayers in a time of economic crisis and giving it to organizations–many of which are anti-Catholic–so they can spend it on killing non-white babies in Third World nations.”)
When I get rebuttals to the points below, I’ll post them here, and we’ll see if the Third Way argument fall apart. Here’s the case presented by Third Way’s Rachel Laser and Nikki Yamashiro:

Repealing the Mexico City Policy: A Life-Affirming Action
U.S. foreign family planning aid is a life-affirming, common ground policy that transcends abortion but has unfortunately gotten caught in the political web of abortion politics.
The “Mexico City” policy, reinstated in 2001, introduced sweeping restrictions for U.S. family planning funding for foreign nongovernmental organizations (NGOs). These restrictions have led to dire health outcomes in the world’s poorest countries, from pregnancy-related deaths to the spread of HIV/AIDS. They have also had the effect of increasing the number of abortions by denying basic access to contraception and thus increasing the rate of unintended pregnancy. Even former President George W. Bush acknowledged problems with the policy when he exempted funding for U.S. HIV/AIDS assistance from the Mexico City policy restrictions.
It is time to disentangle U.S. foreign family planning aid from abortion politics and separate myths from facts. Reforming U.S. foreign family planning aid by repealing the Mexico City policy is a life-affirming step.
Fact: The repeal would not fund abortion.
Since 1973, the Helms Amendment to the Foreign Assistance Act has explicitly banned the use of U.S. taxpayer funds for abortions overseas.4 Repealing the Mexico City policy would do nothing to change this.
Fact: The repeal would reduce the number of abortions worldwide.
A joint report by the Guttmacher Institute and the United Nations Population Fund estimated that providing family planning services to the 201 million women in developing counties whose needs are unmet would prevent 52 million unintended pregnancies and 22 million abortions annually.

• The Planned Parenthood Association of Ghana provided family planning services to as many as 697,000 individuals. Their loss of funding as a result of the Mexico City policy affected the ability of 1,327 communities in Ghana to prevent unintended pregnancies and abortion. Repealing the Mexico City policy would reduce the number of abortions worldwide by restoring desperately needed family planning services to some of the poorest countries in the world.

Fact: The repeal would prevent countless pregnancy-related deaths and illnesses worldwide.
Over 99% of the estimated 536,000 women who die each year from pregnancy-related causes live in developing countries.8 If family planning needs were met for all women in the developing world who do not have access to contraception, pregnancy-related deaths would drop by 25-35%.

• The Mexico City policy has lead to the loss of USAID-supplied contraceptives in 16 developing countries throughout Africa, Asia and the Middle East. Repealing the Mexico City policy would help save the lives of women worldwide by restoring needed family planning services.

Fact: The repeal would save the lives of countless children worldwide.
In developing counties, the timing and spacing of pregnancies plays a significant role in children’s health. For example, in Kenya, if women were able to better space their pregnancies, mortality rates for children under 5 years of age would fall by an estimated 17%.

• As a result of the Mexico City policy, the Family Planning Association of Kenya (FPAK), the oldest and most established family planning NGO in Africa, was forced to close three clinics that had served an estimated 19,000 Kenyans. FPAK clinics, in addition to family planning services, provide prenatal, postnatal, and well baby care. Repealing the Mexico City policy would help save the lives of children worldwide by providing women with family planning services, enabling them to better space their pregnancies and deliver healthy children.

Fact: The repeal would fight the spread of HIV/AIDS and other STIs.
Family planning NGOs play a key role in combating the spread of HIV/AIDS and other STIs in developing countries, saving the lives of countless men, women and children. Repealing the Mexico City policy would restore critical funding for HIV/AIDS and other STI prevention information and services.

• Without intervention and at the current rate of infection, about half of the youth who are now age 15 in Zambia will likely die of AIDS.

The Planned Parenthood Association of Zambia, whose work includes reaching young people with information and services aimed at preventing the transmission of HIV/AIDS, has lost 24% of its funding and almost 40% of its staff as a result of the Mexico City policy.
Fact: The repeal would restore a wide range of health services.
As a result of the Mexico City policy, clinics that provide a number of healthcare services, in addition to family planning services, are struggling and in some cases have closed.

• The loss of USAID funding resulted in the Family Planning Association of Kenya (FPAK) and Marie Stopes International Kenya closing a number of clinics that provided pre- and post-natal care, child immunizations, infant and child check-ups, and malaria screening and treatment. FPAK is also Kenya’s primary provider of Pap smear tests for cervical cancer.

UPDATE: David Gibson agrees with this argument.
UPDATE #2: National Right to Life Committee put out this press release which partly counters the Third Way arguments:

“One effect of Obama’s order will be to divert many millions of dollars away from groups that do not promote abortion, and into the hands of those organizations that are the most aggressive in promoting abortion in developing countries, [says Douglas Johnson]. “President Obama not long ago told the American people that he would support policies to reduce abortions, but today he is effectively guaranteeing more abortions by funding groups that promote abortion as a method of population control.”
Contrary to some misunderstandings, enforcement of the Mexico City Policy did not reduce the amount of money spent on the program, nor will Obama’s order increase the amount (which is $461 million in the current fiscal year). Rather, the policy affects what type of groups qualify for grants under the program. “Obama’s order will predictably result in a redirection of funds to groups such as the International Planned Parenthood Federation, which are ideologically committed to the doctrine that abortion on demand must be universally available as a birth control method,” Johnson said.

Although Obama’s order will result in major subsidies for organizations that promote abortion overseas, the direct use of the U.S. funds to perform abortion procedures will remain unlawful under the Helms Amendment to the Foreign Assistance Act. “The Helms Amendment can be changed only by an act of Congress, but because the Obama Administration is joined at the hip with the abortion lobby, we will be watching carefully for any evidence that the Administration is failing to enforce the Helms Amendment,” Johnson said.
The details of the Mexico City Policy are spelled out in an official handbook issued by the U.S. Agency for International Development (USAID), which is available on request from NRLC (in PDF format). Basically, the policy required grantees to refrain from performing abortions (except to save the life of the mother, or in cases of rape or incest), or lobbying to legalize abortion, or otherwise promoting abortion as a family-planning method. The policy explicitly allowed responding to questions about where abortions may be obtained, in countries in which abortions are legal.