The title to this post is wrong. If you have a "private" insurance, then no Obamacare is not required.
From the site you cited (http://obamacarefacts.com/obamacare-womens-health-services.php), I found the following:
Women across America will benefit from the new provisions and protections in the Affordable Care Act (ObamaCare). As of Aug. 1st, 2012, or at their next renewal date, women will have access to a large number of preventative services which will be completely covered by the insurance companies. These services include:
• Well-woman visits.
• Gestational diabetes screening that helps protect pregnant women from one of the most serious pregnancy-related diseases.
• Domestic and interpersonal violence screening and counseling.
• FDA-approved contraceptive methods, and contraceptive education and counseling.
• Breastfeeding support, supplies, and counseling.
So, according to that site, private insurers will eventually have to provide for breast pumps.